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UNIVERSITY  OF  CINCINNATI. 

Bulletin  No.  13. 


Sekiks  II. 


Publications  of  the  University  of  Cincinnati. 
Edited  by  HOWARD  AYERS. 


Vol..  II. 


LECTURES  ON  THE  THEORY  OF 

MAXIMA  AND  MINIMA  OF  FUNCTIONS 
OF  SEVERAL  VARIABLES. 

(  Weierstrass'  Theory.) 

By  HARRIS  HANCOCK,  Ph.D.  (Berlin),  Dr.  Sc.  (Paris), 
Professor  of  Mathematics. 


Tlie  University  Bulletins  are  Issued  Monthly. 


Entered  at  the  Post  Office  at  Cmcinnati,  Ohio,  as  second-class  matter. 


(iNAn 
1  PR£S& 


o  •■  , 
'-'ft/*.,  p. 


University  of  Cincinnati. 

Bulletin  No,  J  3. 


Publications  of  the  University  of  Gncinnati. 

Series  II.  Vol  II  * 

Edited  by  HOWARD  AVERS. 


I 


i 


LECTURES  ON  THE  THEORY  OF 

'AXIMA  AND  MINIMA  OF  FUNCTIONS 
OF  SEVERAL  VARIABLES. 

(Weierstrass'  Theory.) 

By  HARRIS  HANCOCK,  Ph.D.  (Berlin),   Dr.  Sc.  (Paris),     . 
Professor  of  Mathematics. 


The  University  Bulletins  are  Issued  Monthly. 


Entered  at  the  Port  Office  »i  Cincinnati,  Ohio,  as  sccond-cUss  matter. 


\0 


f# 


PREFACE. 


In  his  lectures  at  Berlin  the  late  Professor  Weierstrass  often 
indicated  the  necessity  of  establishing  fundamental  parts  of  the 
Calculus  upon  a  more  exact  foundation. 

It  has  already  been  pointed  out  {Annals  of  Mathematics,  Vols. 
IX.,  X.,  XI.  and  XII.)  how  the  old  rules  and  theories  of  the  Cal- 
culus of  Va?'iatio7ts  soon  led  to  perplexities  which  appeared 
almost  insurmountable.  Dirichlet's  Principle  is  found  to  have 
been  established  upon  a  weak  structure,  and  we  very  soon  find 
innumerable  fallacies  and  difficulties  when  we  seek  to  discuss  in 
this  manner  Minimal  Surfaces  and  the  allied  theory. 

These  difficulties  may  be  overcome  by  subjecting  the  problems 
in  question  to  a  more  rigorous  treatment  and  by  giving  more 
emphasis  to  their  analytic  formulation. 

In  every  Differential  Calculus  which  I  have  seen  \cf.  also 
Pierpont,  Bull,  of  Amer.  Math.  Soc.,July,  i8()8\  the  Theory  of 
Maxima  and  Minima  is  both  inexact  and  inadequate,  when  several 
variables  are  treated.  This  subject,  when  made  more  rigorous, 
should  evidently  receive  increased  attention.  Indeed,  at  the  pre- 
sent state  of  mathematical  science  it  seems  that  students  should 
devote  more  attention  to  its  study,  for  it  has  a  high  inter- 
est as  a  topic  of  pure  analysis,  and  finds  immediate  applica- 
tion to  almost  every  branch  of  mathematics.  Further,  the  Theory 
of  Maxima  and  Minima  should  receive  more  attention  for  its  own 
sake — for  example,  in  the  solution  of  such  problems  as  the  deter- 
mination of  the  polygon  which,  with  a  given  periphery  and  a 
given  number  of  sides  contains  the  greatest  area,  the  deriva- 
tion of  the  shortest  line  from  a  point  to  a  surface,  etc.  In  the  > 
Calculus  of  Variations  its  use  is  really  essential,  while  in 
Mechanics  it  may  be  shown  that  all  problems  which  arise,  may 

(3) 


175900 


4  PREFACE 

be  reduced  to  problems  of  Maxima  and  Minima;  from  it  we  may 
derive  a  proof  of  the  existence  of  the  roots  of  algebraic  equa- 
tions as  also  a  method  for  the  reversion  of  series. 

I  do  not  assume  credit  for  the  origin  of  any  of  the  theories 
that  are  here  set  forth.  In  the  presentation  of  the  subject-matter 
I  have  followed  Weierstrass'  lectures  delivered  in  the  University 
at  Berlin,  my  lectures  being  for  the  most  part  a  reproduction  of 
his  lectures.  In  these  lectures  Weierstrass  subjected  to  a  more 
rigorous  investigation  the  work  which  is  in  a  great  measure 
due  to  older  writers,  whom  I  have  indicated  in  the  context. 

Before  entering  upon  the  Theory  of  Maxima  and  Minima  it 
seems  advisable  to  give  a  short  account  of  Weierstrass'  Theory  of 
Analytic  Functions  and  to  give  more  exact  definitions  of  those 
functions  to  which  the  ordinary  rules  of  differentiation  are  applic- 
able. This  investigation  is  carried  out  only  so  far  as  the  present 
treatise  seems  to  require.  Certain  theorems  are  also  introduced 
upon  which  the  later  discussion  depends.  The  Theory  of  Maxima 
and  Minima  may  be  then  presented  in  a  clearer  and  more  con- 
nected form. 

My  thanks  are  due  to  Messrs.  Harry  H.  Steinnietz  and  Harold 
P.  Murray  of  the  University  Press  for  their  care  in  the  printing 
of  this  work. 


Harris  Hancock. 


McMicKEN  Hall, 
University  of  Cincinnati. 
Jan.,  1903. 


CONTENTS. 


CHAPTER  I. 


CERTAIN  FUNDAMENTAL  CONCEPTIONS  IN  THE  THEORY  OF 

ANALYTIC  FUNCTIONS. 
ART. 

1  Rational  functions  of  one  or  more  variables.  Functions  defined  through 
arithmetical  operations.  One-value  functions.  Infinite  series  and  in- 
finite products.     Convergence.     Art.  2.     Uniform  Convergence. 

3  Region  of  Convergence.     Differentiation 

4  Many-valued   functions.       Functions   of  several   variables.       Functions 

which  behave  like  an  integral  rational  function 

5  Arithmetical    dependence.       Art.  6.       Many-valued    functions.       Art.  7. 

Possibility  of  expressing  many-valued  functions  through  one-valued 
functions.      ............. 

8  Analytic  functions  expressed  through  power-series.     Analytic  structures. 

9  Analytic  structures  defined  in  another  manner.      Structures  of  the   first 

kind,  second  kind,  etc.         .......... 

10  Analytic  continuation.     Power-series  closed  in  themselves. 

11  Definition  of  analytic  functions.     Function-element.  .... 

12  Existence  of  the  general  analytic  function 

13  Extension  of  the  above  definitions  to  systems  of  functions  of  one  or  more 

variables.     One-valued  functions  defined 

Definition  of  many-valued  functions.     Values  common  to  n  functions. 

14  An  important  theorem  for  the  Calculus  of  Variations.  .... 

15  The  same  theorem  proved  in  a  more  symmetric  manner 

16  Application  of  this   theorem   and   the  definition   of  a   structure  of  the 

(» — wi)th  kind  in  the  realm  of  n  quantities.       .    \ 

17  Property  of  power-series. 

Transformation  of  the  expression  of  a  structure. 

Continuation  of  a  function.     Coincidence  of  two  structures. 

18  A  complete  structure  defined.     Monogenic  structures.  .... 

19  Boundary  positions.  

Singular  systems 

20  More  exact  conception  of  one-valued  and  many-valued  functions. 

21  Points  at  infinity 


PAGE. 


9 
10 


11 


12 
13 

14 
IS 
16 
17 

18 
19 
20 
21 

22 
23 
24 
25 
26 
27 
28 
29 
30 


CHAPTER  II. 


THEORY  OF  MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAL  VARI- 
ABLES THAT  ARE  SUBJECTED  TO  NO  SUBSIDIARY  CONDITIONS. 

1     Introduction.  Art.  2.     Nature  of  the  functions  under  consideration.     Defi- 
nition of  regular  functions, 

3    Definition  of  maxifna  and  minima  of  functions  of  one   and   of   several 


(5) 


31 


6  CONTENTS 

ART.  PACK, 

variables 32 

4  The  problem  of  this  chapter  proposed.     Taylor's  theorem  for  functions 

of  one  variable 33,34 

5  Taylor's  theorem  for  functions  of  several  variables 35,36 

6  The  usual  form  of  the  same  theorem 37,38 

7  A  condition  of  maxima  and  minima  of  such  functions.          ....  39 

THEORY  OK  THE  HOMOGENEOUS  QUADRATIC  FORMS. 

8  Indefinite  and  definite  quadratic  forms.  40 

No  maximum  or  minimum  value  of  the  function  can  enter,  when  the  cor- 
responding- quadratic  form  is  indefinite.  When  is  a  quadratic  form  a 
definite  form  which  only  vanishes  when  all  the  variables  vanish  ?       .  41 

9  Some  properties  of  quadratic  forms.     The   condition   that   the   quadratic 

form   <^  \X\,  X2, Xxi)^^^  A\fiX\Xij,  be  expressed  as  a  function 

of  n — 1  variables. 42-44 

11,12    Kvery  homogeneous  function  of  the  second   degree    ^  (a*i,  X2, A'n  ) 

may  be  expressed  as  an  aggregate  of  squares  of  linear   functions  of  the 

variables 45-47 

10-17    The  question  of  art.  8  answered 44-50 


APPLICATION  OF  THE  THEORY  OF  QUADRATIC  FORMS  TO  THE 
PROBLEM  OP  MAXIMA  AND  MINIMA  STATED  IN  ARTS.    1-6. 

18.     Discussion  of  the  restriction  that  the  definite  quadratic  form  must  only 

vanish  when  all  the  variables  vanish 51 

The  problem  of  this  chapter  completely  solved 52 


CHAPTER  III. 

THEORY  OF  MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  SEVERAI,  VARI- 
ABLES THAT  ARE  SUBJECTED  TO  SUBSIDIARY  CONDITIONS. 

1     The  problem  stated.       Art.  2.     The  natural  way  to  solve  it.         .        .        .  54 
3-6    Derivation  of  a  fundamental  condition  by  the  application  of  the  theorem 

of  arts.  14  and  15  of  Chapter  I. 55-57 

7  Another  method  of  finding  the  same  condition 57 

8  Discussion  of  the  restrictions  that  have  been  made 58 

9  A  geometrical  illustration  of  these  restrictions 59 

10  Establishment  of  certain  criteria. 60 

11  Simplifications  that  may  be  made.  60 

12  More  symmetric  conditions  required 61 

THEORY  OF  HOMOGENEOUS  QUADRATIC  FORMS. 

13  Addition  of  a  subsidiary  condition. 61 

14,  15    Derivation  of  the  fundamental  determinant  Ae  and  the  discussion  of  the 

roots  of  the   equation    Ae^o,    known   as   the   "Equation  of  Secular 

Variations." 61-65 

16    The  roots  of  this  equation  are  all  real.            .......  65 

17-19    Weierstrass'  proof  of  the  above  theorem 66 

20,  21     An  important  lemma. 67,  68 


CONTENTS 


ART. 

22  A  g-eneral  proof  of  a  general  theorem  in  determinants 

23  The  theorem  proved  when  the  variables  are  subjected  to  subsidiary  con- 

ditions. ............. 

24  Conditions  that  the  quadratic  form  be  continuously  positive  or  continu- 

ously negative.  


PAGB. 
69 

70,71 

72 


25 

26 
27 


28 


APPLICATION  OF  THE  CRITERIA  JUST  FOUND  TO  THE  PROB- 
LEM OP  THIS  CHAPTER. 

The  problem  as  stated  in  art.  1  solved  and  formulated  in  a  symmetric 

manner.  .............  73 

The  results  summarized  and  the  general  criteria  stated 73,  74 

Discussion  of  the  geometrical  problem:  Determine  the  greatest  and  the 
smallest  curvature  at  a  regular  point  of  a  surface.  Derivation  of  the 
characteristic  diiferential  equation  of  Minimal  Surfaces.       .        .        •      75-80 

Solution  of  the  geometrical  problem:  From  a  given  point  to  a  given  sur- 
face, draw  a  straight  line  whose  length  is  a  ininimwin  or  a  maximum.      80-84 


CHAPTER  IV. 


7 
8 
9 

10 


11-16 
11 


SPECIAL  CASES. 

THE  PRACTICAL  APPLICATION  OF  THE  CRITERIA  THAT  HAVE  BEEN 

HITHERTO  GIVEN  AND  A  METHOD  FOUNDED  UPON  THE 

THEORY  OF  FUNCTIONS,  WHICH  OFTEN  RENDERS 

UNNECESSARY  THESE  CRITERIA. 

Difficulties  that  are  often  experienced.  85 

Fallacies  by  which  maxima  and  minima  are  established,  when  no  such 

maxima  or  minima  exist.  86 

Definitions:     Realm.     A  position.     An  n-ple  multiplicity.     Structures.     A 
position  defined  which  lies  within  the  interior  of  a  definite  realm,  on 

the  boundary,  or  without  this  realm. •    .        .  87 

Statement  of  two  important  theorems  in  the  Theory  of  Functions.     .        .      87,  88 
Upper  and  lower  limits  for  the  values  of  a  function.   Asymptolic  approach. 

Geometrical  and  graphical  illustrations. 88,  89 

Cases  where  there  are  an  infinite  number  of  positions  on  which  a  function 

may  have  a  maximum  value 89,  90 

Reduction  of  such  cases  to  the  theory  of  maxima  and  minima  proper.         .      90,  91 
The  derivatives  of  the  first  order  must  vanish.       ..."..  91 

The  results  that  occur  here  are  just  the  condition  which  made  the  former 

criteria  impossible 91 

The  previous  investigations  illustrated  by  the  problem:     Among  all  poly- 
gons which  have  a  given  number  of  sides  and  a  given  peritneter,  find 

the  one  which  contains  the  greatest  surface-area 92 

Solution  of  the  above  problem. 92-97 

Cremona's  criterion  as  to  whether  a  polygon  has  been  described  in  the 

positive  or  negative  direction.  93 


CASES  IN  WHICH  THE  SUBSIDIARY  CONDITIONS  ARE  NOT   TO 
BE  REGARDED  AS  EQUATIONS  BUT  AS  LIMITATIONS. 

17    Examples  illustrating  the  nature  of  the  problem  when  the  variables  of  a 
given  function  cannot  exceed  definite  limits. 


97 


8  CONTENTS 

ART.  PAGB. 

18  Reduction  of  two  inequalities  to  one ,         .  <»      97 

19  Inequalities  expressed  in  the  form  of  equations.             98 

20  Examples  taken  from  mechanics.              98 

gauss'  principle. 

21  Statement  of  this  Principle.  99 

22  Its  analytical  formulation 100 

23  By  means  of  this  Principle  all  problems  of  mechanics  may  be  reduced  to 

problems  of  maxima  and  minima.  101 

TWO  APPLICATIONS  OF  THE  THEOKY  OF  MAXIMA  AND  MINIMA  TO 
REALMS  THAT  SEEM  DISTANT  FROM  IT. 

/.     Cauchy's  prooj  of  the  existence  of  the  roots  of  algebraic  equations. 

24  Statement  of  the  proof  of  this  theorem. 101,  102 

//.    Proof  of  a  theorem  in  the  Theory  of  Functions.    Reversion  of  Series. 

25  Statement  of  the  theorem 102 

26  Recapitulation  of  what  has  been  done  in  the  previous  investigation  re- 

g-arding  the  reversion  of  series. 102 

27  The  theorem  in  its  modified  form  is  thus  stated:     (1)   When  the  variables 

X  and  y  are  connected  by  the  equations 

li=n 


[1]  :>'x=2(^^''+  A^m)^^' 


M=l 


where  dXfi  and    \  ,     are  quantities  which  are  subjected  to  certain 

conditions,  then  it  is  always  possible  to  fix  for  the  variables   X^,  X2,  •  • 

.  .X^   and  j^-i,  y^, j>/„   definite  limits  g^,  g^, g^   and 

hi,  hi,  .  .  .  .  h„    in  such  a  manner  that  for  every  system  of  the  y'sfor 

which   \y\  I  <^  h\  (A.=l.  2, «)  there  exists  one  system  of  the  x'sfor 

which   |;i;x|<^^x  (\=1,  2, n)  so  that  \\]  is  satisfied.     (2)     Thesolu- 

tion  of  [1]  has  a  similar  form  as  the  equations  [1]  themselves,   viz: — 


X\  = 


2   (^Xm +Yxm  ).>''' • 


(X=l,  2, fi)     .      ■      .103,104 

28-36     Proof  and  discussion  of  this  theorem.     Determination  of  upper  and  lower 

limits  for  the  quantities  that  occur. 105-114 

32  Unique  determination  of  the  system  of  values  of  the  .r's  that  satisfy  the 

equations  [1]  above.  108 

33  If  the  j/'s  become  infinitely  small  with  the  .ar's,  the  .r's  become  infinitely 

small  with  the  jy's.     The  jr's  are  continuous  functions  of  the  jy's.        .  109-111 

34  The  x's,,  considered  as  functions  of  thejv's,  have  derivatives  which  are  con- 

tinuous functions  of  the  jv's.     The  existence  of  the  first  derivatives.    .  111-113 

35  The  .r's,  expressed  in  terms  of  the  jt/'s,  are  of  the  same  form  as  the  given 

equations  expressing  the  jv's  in  terms  of  the  .^''s.  ....  113 

36  Conditions  which  must  exist  before  the  ordinary  rules  of  differentiation 

are  allowable  in  the  most  elementary  cases.  113,  114 


CHAPTE)R  I. 

CERTAIN  FUNDAMENTAL  CONCEPTIONS  IN  THE  THEORY  OB' 
ANALYTIC   FUNCTIONS. 

1.  In  the  development  of  the  conception  of  the  analytic  func- 
tions if  we  start  with  the  simplest  functions  which  may  be  expressed 
through  arithmetical  operations,  we  come  first  to  the  rational 

/unctions  of  one  or  more  variables.  The  conception  of  these 
rational  functions  may  be  easily  extended  by  substituting  in  their 
places  one-valued  functions,  and  first  of  all  those  which  may  be 
again  expressed  through  arithmetical  operations,  viz. — sums  of  an 
infinite  number  of  terms  of  which  each  is  a  rational  function,  or 
products  of  an  infinite  number  of  such  functions. 

The  necessity  at  once  arises  of  developing  the  conditions  of 
convergence  of  infinite  series  and  products,  since  such  an  arith- 
metical expression  represents  a  definite  function  only  for  values 
of  the  variables  for  which  it  converges.  Mere  convergence,  how- 
ever, is  no't  sufficient,  if  we  wish  to  retain  for  the  functions  just 
mentioned  the  properties  which  belong  to  the  rational  and  the 
ordinary  transcendental  functions.  All  such  functions  have  de- 
rivatives, and  in  order  to  have  this  property  the  above  expres- 
sions of  one  variable  must  converge  uniformly  (gleichmassig)  in 
the  neighborhood  of  each  definite  value. 

2.  When  we  say  a  function  of  one  variable  converges  uni- 
fortnly,  we  mean  the  following:*  It  is  assumed  that  the  function 
in  question  has  a  definite  value  for  x  —  x^.  We  next  consider 
all  values  of  x  for  which  x  —  x^^  does  not  exceed  a  definite  quantity 
d.  We  shall  then  suppose  that  the  series  is  to  be  convergent  for 
a  given  value  of  x  that  lies  within  this  interval.  In  order  that 
this  series  converge  uniformly,  it  must  be  possible,  after  we  have 


*A  different  definition  is  given  by  Weierstrass  (Collected  works,  vol.  II.,  p.  202  and 
Zur  Functionenlehre,  \\.) 

(9) 


10  Theory  of  Maxima  and  Minima 

assumed  an  arbitrarily  positive  quantity  8  and  when  a  remainder 
R„  {pc)  has  been  separated  from  the  series,  to  find  a  positive  in- 
teger tn  so  that 

I  ^  n  (^)  i  -^  ^  >  where  n  >  tn 

for  all  values  of  x  in  this  interval.  \^Cf.  Dini,  Theorie  der  Func- 
tionen,p.  ijy.  Translat^dinto  German  by  Lilroth  and Schepp^ 
3.  The  conditions  of  uniform  convergence  being  retained,  it  is 
essential  that  all  the  transcendental  functions  have  a  property  in 
common  :  If  we  take  a  value  x^  within  the  region  of  convergence 
(Convergenzbezirk)  in  which  these  functions  considered  as  func- 
tions of  one  variable  converge  uniformly,  then  they  may  be  repre- 
sented for  all  the  values  of  x  in  the  neighborhood  of  x^  as  series 
which  proceed  according  to  positive  integral  powers  of  x — x^. 
For  example, 

/(^)=/(^— ^o  +  ^o)=^o+«i  {x—x^^a^  (x—XoY  + , 


where  ao,  «i,  a2, are  definite  functions  of  Xq. 

From  this  it  follows  that  they  may  be  differentiated  and  a  number 
of  other  properties  are  immediate  consequences. 

4.  Starting  from  the  same  standpoint  as  in  the  case  of  one- 
valued  functions  we  may  in  a  similar  manner  give  to  many-valued 
functions  a  definition  which  is  far  reaching  in  its  generality.  For 
this  purpose  the  conception  of  the  usual  operations  must  be  ex- 
tended to  several  variables.  It  is  then  easy  to  extend  in  the 
required  manner  the  conception  of  uniform  convergence. 

Let  (21,  (22' <^n  be  a  definite  system  of  values  of  the  variables 

x^,  X2, .  . .  .  x„  within  the  region  of  uniform  convergence,  and  con- 
sider only  the  values  of  x^,  X2, . . .  .  x^  for  which  Xi  —  a^,  X2—a2, .... 
^n— <3^ii  do  not  exceed  certain  limits  d^,  ^2>  •  •  •  -^n-  l^he  function 
may  be  then  represented  through  an  ordinary  series  which  pro- 
ceeds according  to  integral  powers  of  Xi—ai,Xi — a2,X2—a2,.-. 
Xn—a„,  and  consequently  may  be  differentiated;  in  short,  it  behaves 
— as  Weierstrass  was  accustomed  to  express  it — like  an  integral 
rational  function  in  the  neighborhood  of  a  definite  position 
within  the  interior  of  the  region  of  uniform  convergence. 

5.  The  definition  of  arithmetical  functions  being  established 
as  in  the  preceding  articles,  the  conception  0/  arithmetical  de- 


ER31TY 


of  Functions  of  Several  Variables. 


11 


pendence  among  several  variables  may  be  defined  as  follows: 
If  we  represent  a  function  which  has  been  formed  as  indicated 
above  by  i^(;ti,  Xi,...x^,  then  F(xi,  Xj, . .  Xa)—0  expresses  a 
certain  dependence  among  the  variables  x^,  X2,  .  ■  .  x„;  that  is, 
among  the  infinite  number  of  systems  of  values  for  which  the 
function  has  a  meaning,  those  which  satisfy  this  equation,  are  to 
be  taken.  There  exists,  therefore,  among  x^,  X2, .  . .  x„  a.  depend- 
ence of  a  similar  character,  as  in  the  case  of  algebraic  equations. 
If  we  choose  the  quantities  Xi,  x^,  .  ■  ■  x^  such  that  the  equations 
Fi=o,  F-i,~o,. . .  .  F^=-.o,  where  nt<n,  are  true,  then  we  have  a 
dependence  among  the  quantities  Xi,  x^,.  • .  x^  defined  in  such  a  way 
that  at  all  events  we  can  choose  arbitrarily  not  more  than  n  —  m 
of  the  variables,  since  the  remaining  'fn  variables  are  determined. 

6.  In  conformity  with  what  was  given  in  articles  4  and  5  we 
ma}''  develop  also  the  conception  of  many  valued  functions. 

Suppose,  for  example,  a  function  of  two  variables  x  and  jk  is 
given;  then  we  may  consider  all  the  systems  of  values  {x,  y)  in 
which  X  has  a  prescribed  value.  For  such  a  value  of  x,  several 
values  of  y  may  be  found.  We  consider,  then,  j^-  as  a  function 
of  X,  and  this  function  will  be  a  many-valued  function. 

The  fact  that  by  proceeding  in  this  way  we  cannot  grasp  all 
the  possible  different  forms  of  function  has  given  occasion  to 
present  the  conception  of  analytic  functions  in  a  different  manner 
in  that  we  make  a  characteristic  property  for  the  foundation 
of  the  definition. 

Weierstrass  wished  in  no  wise  to  reject  the  mode  of  concep- 
tion that  has  been  used  up  to  this  point,  in  accordance  with 
which  we  have  to  define  every  analytic  dependence  through 
quantities  that  may  in  reality  be  expressed  in  an  arithmetical 
manner;  for  it  will,  in  all  probability,  finally  result  that,  how- 
ever the  analytic  function  may  be  defined,  every  dependence  may 
be  represented  in  the  form  indicated. 

7.  The  following  considerations  will  serve  to  explain  the 
remarks  of  the  previous  article.  Cases  of  dependence  have  been 
known  for  a  long  time  where  it  could  not  be  proved  that  they 
could  be  represented  in  an  arithmetical  manner.  As  an  example 
let  any  analytic  dependence  exist  between  two  variables  and  limit 
one  of  the  variables  x  \.o  a.  definite  region;    then  the  other  varia- 


12  Theory  of  Maxima  and  Minima 

ble  y  may  be  represented  through  x.  We  wish  to  express  this  rep- 
resentation in  a  form  that  remains  continuously  true.  If  to  a 
value  of  the  one  variable  a  transcendental  function  of  the 
other  variable  corresponds,  we  do  not  see  the  possibility  of 
expressing  one  variable  arithmetically  in  terms  of  the  other 
variable. 

Many  examples  are  known  in  which  the  idea  is  conceived  of 
expressing  the  variables  x  and  y  in  terms  of  a  third  variable,  and 
thereby  representing  both  of  the  original  variables  as  one-valued 
functions  of  a  third  variable  in  such  a  way  that  if  we  give  to 
this  variable  all  possible  values  we  have  all  systems  of  values 
{x,  y).     The  simplest  example  is  perhaps  the  following: 

If  we  take  the  equation  z^x'',  where  x  and  y  are  to  be 
two  independent  variables,  then  it  is  in  no  wise  possible  to 
express  the  above  dependence  in  an  arithmetic  form;  that  is,  in  a 
form  in  which  transcendental  functions  do  not  appear.  But  if  we 
introduce  a  third  variable  /,  and  write  x=e\  then  we  have  x^'=e^'\ 
and  consequently  x=^e*,  z=e^'\ 

Thus  X  and  y  are  expressed  in  one-valued  form  by  the  help 
of  t. 

Much  use  has  already  been  made  of  such  dependence ;  for 
example,  in  the  differential  equation  of  the  hypergeometric  series. 

This  differential  equation  has  among  others  two  independent 
solutions  which  are  transcendental  functions  of  x.  It  was  not  at 
first  known  how  to  express  this  connection,  but  it  was  afterwards 
found  that  if  we  exclude  from  the  values  which  the  variable  x  can 
take  those  values  which  belong  to  a  definite  line  (namely,  the  real 
values  from  + 1  to  +  co ),  then  for  all  remaining  values  the  series 
may  be  expressed  through  a  sum  of  rational  functions. 

Professor  Poincare  has  shown:*  If  between  x  and  y  an  al- 
gebraic equation  exists,  then  all  the  systems  of  values  may  be 
expressed  in  the  form,  indicated.  He  next  showed  that  the 
same  could  be  effected  for  the  linear  differential  equations  by 
the  help  of  functions,  to  which  Fuchs  had  first  called  attention,  and 


*  See  Bulletin  de  la  Soci^t6  math^matique  de  France  (t.  XI,  1883). 
See  also  lectures  II  and  III,  delivered  in  the  Cambridg-e  Colloquium,  by  Professor 
Osgood  [Bulletin  of  the  Amer.  Math.  Society,  1898]. 


of  Functions  of  Several  Variables. 


13 


which  Poincare  calls  " Fuchsian  Functions."  If  we  have  any 
linear  dififerential  equation,  then  we  may  express  the  two  varia- 
bles X,  y  as  one-valued  functions  of  a  third  variable. 

The  functions  which  have  been  introduced  by  Poincar^  are 
extremely  useful  when  we  have  to  do  with  the  linear  dififerential 
equation  of  the  second  order.  Poincare  found  finally  the  fol- 
lowing general  theorem:  if  in  general  there  exists  an  analytic 
dependence  between  two  variables  x  and  y,  it  is  always  pos- 
sible to  represent  x  and  y  as  one-valued  functions  of  a  third 
variable.  In  this  manner  the  study  of  many-valued  functions 
is  reduced  to  the  study  of  one-valued  functions.  However  this 
is  not  a  simple  task,  for  if  we  wish  in  reality  to  make  this  rep- 
resentation in  the  case  of  a  linear  differential  equation,  we 
encounter  many  technical  difficulties.  Nevertheless  it  is  essential 
to  prove  that  there  exist  such  representations. 

Further  progress  has  not  been  made  in  this  direction,  but 
Weierstrass*  thought  that  this  end  could  be  attained  for  the 
theory  of  functions,  viz.:  that  it  is  always  possible  where  an 
analytic  dependence  exists  between  two  variables  to  express 
this  dependence  in  a  one-valued  form  which  remains  continuously 
true.  In  this  treatment  one  may  use,  besides  the  variables,  still 
other  auxiliary  variables. 

8.  After  what  has  been  given  above  it  seems  advisable  to 
follow  a  different  method  than  the  one  hitherto  employed,  so 
that  we  may  come  to  a  more  comprehensive  definition  of  the 
analytic  functions,  namely :  a  method  indicated  by  a  charac- 
teristic property  of  all  the  functions  under  consideration,  the 
representation  through  power- series  for  limited  values  of  the 
variables. 

Suppose  first  an  analytical  relation  exists  between  two  varia- 
bles; then  this  may  be  expressed  through  an  analytic  equation 
between  x  and  y  of  the  formf 

P{x—x^,y—y^:=o, 
if  (;v<„  jv„)  is  a  definite  pair  of  values  of  the  variables.     It  is  thus 

*Weierstrass  made  this  assertion  in  May,  1884. 
See  also  Problimes  MatMntatiques  of  Professor  Hilbert  in  the  Cotnptes  rendus  of  the 
Congress  of  Mathematicians,  Paris,  1900.   • 

t  The  letter  P  has  here,  as  throughout  this  discussion,  simply  the  meaning  "power 
series." 


14  Theory  of  Maxhna  and  Minima 

seen  that  in  the  neighborhood  of  (^Pq.  y^  there  is  an  infinite 
number  of  systems  of  values  which  satisfy  the  equation.  The 
collectivity  of  these  pairs  of  values  {^x,  y)  is  called  an  analytic 
structure — or  configuration  (Gebilde) — in  the  realm  (Gebiet)  of 
the  quantities  x,  y. 

As  in  the  case  of  two  variables,  we  shall  proceed  in  a  similar 
manner  with  several  variables,  among  which  an  analytic  connec- 
tion exists  that  is  expressible  in  arithmetical  form.  Let  this 
connection  be  of  such  a  nature  that  '>n{<n)  of  the  variables 
are  in  general  determined  through  the  remaining  n—m.  If, 
then,  (^1,  a-i, .  ■  ■  a„)  represents  a  definite  system  of  values  of  the 
variables,  there  exist  m  equations  of  the  form 

P{x^—a^,  x^—a^, ....  x,—a„)  =o, 

which  are  to  be  satisfied   for  x^^^a^,  X2=a2, ;r„=a„.      In  the 

neighborhood  of  the  position  {a^,  a^,,  . .  a„)  there  are,  then,  an 
infinite  number  of  other  systems  of  values  (xi,  X2,  ■• .  x„)  which 
satisfy  the  same  m  equations.  These  define  an  analytic  struct- 
ure in  the  realm  of  the  quantities  Xi,  X2, ....  x^.  (Cf.  Chap.  IV,' 
Art.  3.). 

9.  Analytical  structures,  as  above  defined,  may  be  repre- 
sented in  a  different  manner.  Since  the  equation  between  x  and  y 
begins  with  terms  of  the  first  dimension  we  may  either,  on  the 
one  hand  express  y—yo  through  P{x—Xo)  or  x — Xo  through 
P{y—yo),  or,  on  the  other  hand,  if  the  coefficient  of  x—Xq,  or 
y—yo  is  equal  to  zero,  it  is  possible  to  express  only  y—yo,  or  only 
x—Xo  as  integral  power-series  of  x—Xo  or  y—yo.  In  order  that  this 
distinction  be  not  necessary,  we  take  any  function  /  which  begins 
with  terms  of  the  first  dimension  in  x-Xq,  y—yo  {see  Art.  15); 
we  may  then  always  express  the  two  quantities  x^y  as  power-series 
of  t.  Through  the  assumption  of  such  an  equation  it  is  always 
possible  to  include  within  certain  limits  all  the  systems  of  values 
{x—Xo,  y—yo)  which  satisfy  this  equation.  These  values  must 
firstly,  satisfy  the  given  equation,  and  secondly,  afford  all  the 
systems  of  values  which  satisfy  it. 

These  considerations  may  be  extended  at  once  to  equations 
between  several  variables.  If  we  "have  a  certain  number  of  equa- 
tions between  x-^—a<^,  Xj—a^, . . .  x^ — a,„  and  if  we  limit  these  equa- 


of  Functions  of  Several  Variables.  15 

tions  to  terms  of  the  first  dimension,  we  have  linear  homogeneous 
equations  of  the  first  dimension  the  number  of  which  we  assume 
to  be  ^w<«. 

If  we  can  express  m  of  the  quantities  x^—a-^,  X2—CI21  —  ^n—^xi 
through  the  remaining  n—m,  it  is  always  possible  so  to  de- 
rive n  power  series  of  the  n—  m  quantities  t■^,  t^, 4-m  that 

they,  substituted  for  x^,  x^, ^„,  firstly,  satisfy  the  given  equa- 
tions, and  secondly  they,  if  we  give  to  t^,  h,  ...  4-m all  possible 
values,  afford  all  the  systems  of  values  {x-^,  x^,  .  .  x„)  which  sat- 
isfy those  equations,  when  certain  limits  are  fixed  for  the  absolute 
values  of  x^—a^,  x^—a-i,  •  •  x,^ — a^,  or  also  secondly,  that  with  infi- 
nitely small  values  of  the  ^'s  they  afford  all  the  systems  of  values 
of  the  quantities  x^,  X2,...  x„  which  lie  infinitely  near  the  position 
{ui,  a^, .  ■  ■ .  a^)  [see  again  Art.  15]. 

If  now,  reciprocally,  we  assume  n  power-series  of  t-^,  tj,...  4-ni 
and  write  these  equal  to  x^,  x^, x^,  the  collectivity  of  the  sys- 
tems of  values  {^x-^,  X2, x„)   offered   through  these  equations 

constitute  a  structure  of  the  {n — mY^  kind  (Stufe)  in  the  realm 
of  the  quantities  Xi,  X2, ... x„. 

Take  n  power-series  *i  (^),  *2  (0' *«  (l)  ^J^d  write  Xi=^^i{t), 

X2=%(t), Xn=^a{l)'^  then  through  these  equations  a  structure 

of  the  first  kind  in  the  realm  of  the  n  quantities  x  is  defined; 
in  a  similar  manner  a  structure  of  the  second  kind  is  defined 
through  the  equations 

^l=*l  (A,  A),  ^2=*2(A,  ^2). ^n=*n  ( ^^l,    ^2),   CtC. 

10.  The  analytic  structures  that  were  characterized  in  arti- 
cles 8  and  9  have  a  very  important  property,  viz.,  that  the.y  may 
be  continued  (fortgesetzt). 

The  conception  of  continuation  belongs  to  the  most  import- 
ant conceptions  of  the  more  recent  theory  of  functions.  If  instead 
of  a  we  consider  a  position  a^  in  the  region  of  convergence  of  the 
series  that  defines  the  analytic  function,  then  we  may  resolve  this 
series  that  proceeds  according  to  powers  of  x — a  into  a  series 
that  proceeds  according  to  powers  of  x — a-^.  This  series  has, 
then,  another  region  of  convergence  besides  the  one  originally  given; 
in  general,  the  two  regions  of  convergence  will  be  partly  separated, 
so  that  consequently  the  latter  series  has,  at  certain  positions, 
a  meaning  at  which  the  original  series  no  longer  had  a  meaning. 
They  are  identical  for  all  values  of  x  which  lie  in  the  neighbor- 


•  16  Theory  of  Maxima  and  Minima 

hood  of  «!.  On  the  other  hand  it  is  easy  to  show  that,  if  a  definite 
property  belongs  to  one-power  series  (for  example,  if  it  satisfies 
a  definite  differential  equation)  this  property  belongs  to  all  the 
power-series  within  the  whole  region  of  convergence.  We  are 
therefore  compelled  to  look  upon  the  second  power-series  as  a 
continuation  of  the  first. 

In  this  way  we  may  derive  an  indefinite  number  of  power- 
series  from  the  first  one,  and  it  is  easy  to  see  that  if  such  a 
power-series  possesses  one  or  more  definite  properties,  which  may 
be  expressed  in  the  form  of  equations,  the  same  properties  belong 
also  to  each  of  the  continuations. 

It  may  be  further  shown  that  if  a  power-series  P{^x — a^  can 
be  derived  from  another  power-series  Pi^x — a),  then  this  prop- 
erty of  being  derived  is  true  also  when  performed  backwards, 
either  directly  or  also  by  the  aid  of  other  power-series.  It  is 
also  clear  that  if  we  consider  two  power-series  in  x—a ' ,  x — a " 
which  are  derived  from  a  power-series  x — a,  these  series  may  be 
derived  the  one  from  the  other.  The  collectivity  of  the  power- 
series  which  may  be  derived  from  one  single  power-series  as  indi- 
cated, forms,  in  a  certain  measure,  a  whole  which  is  closed  in 
itself. 

11.  According  to  the  previous  article  we  shall  define  an 
analytic  function  of  one  variable  as  follows :  Consider  a  power- 
series  of  X  assumed  or  given  in  any  manner;  let  x'  be  a  defi- 
nite value  of  the  variable  x.     Then  three  things  may  happen: 

(1)  x'  m-ay  lie  in  the  region  of  convergence  of  a  series  that 
is  derived  from  the  given  series;  the  value  for  x=x'  of  this 
series  is  a  value  of  the  analytic  function  which  is  determined 
through  the  original  series.  In  other  words  :  If  with  Weier- 
strass  we  call  the  original  series  and  every  series  derived 
from,  this  one  with  regard  to  the  function  which  they  repre- 
sent, a  FUNCTION-ELEMENT  (Functionenelement)^  then  the  first 
possibility  consists  in  that,  if  any  PUNCTION-ELEMENT  is  given, 
the  definite  value  x'  lies  in  the  region  of  convergence  of  a  func- 
tion-element which  is  derived  from  the  given  one. 

(2)  It  may  happen  that  x  does  not  lie  in  the  region  of 
convergence  of  any  series  that  has  been  derived  in  this  m-anner 
and  that  also  we  cannot  derive  from  the  original  function- 
element  another  function-element  whose  region  of  convergence 


of  Functions  of  Several  Variables. 


17 


can  come  as  near  to  the  point  x'  as  we  wish.  In  this  case  the 
function  does  not  exist  for  x^x' . 

(3)  Although  we  cannot  find  a  power-series  within  which 
x'  lies,  nevertheless,  it  sometitnes  happens  that  we  may  still 
derive  elements  whose  regions  of  convergence  contain  positions 
which  can  come  as  near  to  the  point  x  as  we  wish.*  Whether 
we  can  then  define  the  function  for  x=x'  by  the  considera- 
tion of  boundary  conditions  must  in  each  case  be  considered  for 
itself. 

If  we  have  case  (1)  before  us,  then  the  function  is  defined 
not  only  for  every  value  x'  but  also  for  all  values  in  the  neigh- 
borhood of  X'  and  has  for  these  values  the  character  of  an  in- 
tegral function. 

12.  The  definition  of  an  analytic  function  given  in  the  pre- 
ceding article  is  preferable  to  other  definitions  from  the  fact  that 
the  existence  of  general  analytic  functions  is  at  once  recognized; 
in  short,  that  we  have  in  our  power,  in  our  possession,  all  possible 
analytic  functions.  Kvery  possible  power-series  within  a  region 
of  convergence  gives  rise  to  the  existence  of  a  definite  analytic 
function.  Moreover,  one  must  assume  the  duty  of  proving  in  the 
case  of  every  example,  that  it  leads  to  just  such  functions. 

For  this  reason  investigations  are  necessary  of  which  in  the 
olden  times  we  find  no  trace.  If  we  have  a  differential  equation, 
we  must  begin  with  the  proof  that  the  functions  which  satisfy 
the  differential  equation  arise  from  such  function-elements  that 
were  explained  in  article  11;  therefore,  we  must  first  show  if  y 
is  the  unknown  function  and  x  is  the  variable  of  the  differen- 
tial equation,  that  this  equation  can  be  satisfied  through  y  = 
P{x — a).  Reciprocally,  we  must  show  that,  if  any  variable  quan- 
tity y  is  so  connected  with  another  variable  quantity  x  that  it 
satisfies  the  differential  equation,  then  it  may  be  derived  from 
one  single  function-element  in  the  manner  indicated.  This  last 
proof  is  of  especial  importance  in  the  application  of  analysis  to 
geometrical  mechanics. 

When  a  problem  is  given  in  mechanics^  we  have  to  represent 
the  coordinates  of  the  moving  point  as  functions  of  the  time. 
Only  real  values  are  permitted  in  this  representation.     We  can 


Cf.  Chap.  IV,  Art.  3. 


18  Theory  of  Maxima  and  Minima 

not  therefore  a  priori  know  whether  the  required  functions  are 
analytic  or  not. 

These  functions  are  generally  defined  through  differential 
equations.  We  shall  give  the  simplest  case  as  an  example.  Sup- 
pose we  have  a  system  of  points  that  attract  each  other  according 
to  an  analytic  law,  and  let  x^,  x^,  . .  .  x„  be  the  coordinates  of  these 
points.  If  the  motion  is  a  free  one,  we  have  the  differential 
equation  in  the  form 

d^  Xi 


dt^ 


+ —  -''   \^\i    ^2  1 ^n  j> 


where  F  denotes  a  given  function  of  Xi,  X2, . . . .  x^.  With  such  a 
problem  we  have  to  prove  before  everything  else  that  the  required 
functions  of  time  are  analytic  functions.  If  for  the  point  /=4 
the  initial  position  and  the  initial  velocity  are  given,  then  in  the 
neighborhood  of  the  initial  position  we  can  find  power-series,  and 
we  have  to  show  that  through  these  power-series  the  required 
functions  are  completely  determined. 

13.  If  we  start  with  the  definition  of  a  function  given  in 
Art.  11,  and  if  we  have  for  a  definite  value  x'  a  definite  value  of 
the  function,  then  this  value  depends  essentially  upon  the  way  and 
manner  how  we  come  to  x',  that  is,  upon  the  choice  of  the  values 
Ui,  a2,  .  .  .  Un,  by  means  of  which  we  have  come  from  one  element 
to  the  derived  element  in  question.  From  this  it  may  be  con- 
jectured that  one  and  the  same  function  can  have  different  values 
for  the  same  values  of  the  argument;  these  values  can  not  be 
regarded  as  distinct  from  one  another,  but  are  to  be  considered  as 
vahies  of  the  function. 

If  a,  lies  within  the  region  of  convergence  of  the  first  series, 
a^  in  that  of  the  second,  ....  a„  in  that  of  the  n^^  series,  then  we 
have  a  definite  power-series  P(^x — a^.  If  now  we  take  instead  of 
«i,  ^2, . . .  .  a„_i,  n — 1  other  values  a'l,  a'2, ....  a'„-i,  then  we  can 
have  another  power-series  P,  {x — a^,  so  that  the  function  for 
every  value  x  in  the  region  of  convergence  of  this  series  has  a 
different  value  than  in  the  region  P  {^x — a„). 

Accordingly^  we  may  offer  the  following  definition:  A  func- 
tion is  called  one-valued  on  the  position  a  ,  if  in  the  neighborhood 
of  d  we  may  derive  a  function-element  from  the  original  function- 
element  and  always  have  only  one  function-element  along  what- 


of  Functions  of  Several  Variables. 


19 


ever  path  we  have  come  from  a  to  a'.  If  we  have  several 
function-elements,  then  the  function  is  said  to  be  many -valued, • 
it  may  be  infinitely  many-valued. 

Suppose  next  we  have  a  system  of  functions  of  one  variable 
X.  Let  n  function-elements  of  x  be  given.  Each  of  the  ele- 
ments determines  an  analytic  function.  It  must  be  shown  how 
the  values  of  these  n  elements  are  arranged  with  respect  to  one 
another.  If  the  functions  are  all  or  in  part  many-valued,  the 
question  arises  how  are  we  to  arrange  them  with  respect  to  their 
different  values  and  with  respect  to  one  another.  If,  for  example, 
we  have  two  algebraic  equations  between  the  quantities  x  and  y^ 
we  have  through  the  elimination  of  y  an  equation  for  the  deter- 
mination of  the  possible  values  of  x,  and  by  analogy  we  may 
through  the  elimination  of  x  form  an  equation  for  y.  We  can- 
not write  down  an  arbitrary  value  of  y  for  every  x,  but  in  general 
for  each  of  the  possible  values  of  y  we  can  take  only  one  value 
of  X.  • 

This  investigation  is  very  simple  in  the  case  before  us :  We 
have  values  belonging  in  canmon  to  n  functions  that  are  defined 
in  the  given  manner,  if  we  always  make  use  of  the  same  interme- 
diary a's  for  the  determination  of  the  functions.  In  order  to 
have  a  system  of  values  belonging  in  common  to  the  functions,  for 
x^^x' ,  we  take  a'  in  such  a  manner  that  it  lies  within  the  region 
of  convergence  of  all  the  function-elements,  the  latter  becoming 
P{x — a').  In  the  same  way  we  take  a  quantity  a"  which  lies 
within  the  region  of  convergence  of  all  the  function-elements 
P{^x — a').  These  become  P{^x — «").  Continuing  in  this  man- 
ner we  have  a  system  of  values  belonging  in  common  to  tHe 
function. 

In  algebra  and  in  all  branches  of  mathematics,  where  the 
connection  of  functions  is  defined  through  equations,  it  must  be 
shown  that  we  have  in  reality  by  this  procedure  the  values  which 
belong  in  common  to  the  functions. 

These  definitions  may  be  extended  to  functions  of  several 
variables  and  to  systems  of  functions  of  several  variables. 

We  must  )'et  show  how  functions  of  one  variable  are  to  be 
defined  for  the  boundaries.  This  inve.stigation  will  be  reserved, 
however,  for  another  occasion  (Art.  19),  where  we  shall  present 
the  conception  of  an  analytic  function  in  a  different  manner. 


20 


Theory  of  Maxima  and  Minima 


14.  We  prove  next  a  theorem  extensively  used  in  the  Calculus 
of  Variations. 

Suppose  that  between  the  variables  x^,  x-^,. . .  x^  we  have  m 
equations  given  which  may  be  represented  in  the  form  of  power - 
series,  and  let  these  be: 

c,.i  (;t,— ai)  +  ....  +Ci,„  (;p„— a„)  + V  =o, 
Cu  (^1— «i)  + .  •  •  •  +  C2.„  (^„— a„)  +  V  =0, 

Cm.l(^l—«l )+....   +<:^m.n(^n— «u)  +  Y     =0, 

where  V  ,  V  ,  —  V     are    also    power-series   of  x^ — ^i, 

—  x^ — a„,  but  of  such  a  nature  that  each  term  in  them 
is  of  a  higher  dimension  than  the  first. 

The  equations  will  be  satisfied  for  x^ — ai, ;?;„=«„.      We 

propose   the  problem   of  determining  all  systems   of  values 

{xi,  X2, x^,  which  lie  in  the  neighborhood  ofi^a-^,  a^,  ■  ■ .  .a,), 

and  which  satisfy  the  m-  equations  above;  that  is,  among  the 

systems  of  values  for  which  \  x^ — a,  |  ^ |  x^^ — a„  |  are  smaller 

than  a  fixed  limit  p,  determine  those  which  satisfy  our  m, 
equations. 

The  quantity  p  is  only  subject  to  the  condition  of  being  suffi- 
ciently small.  To  solve  this  problem  we  consider  the  system  of 
linear  equations,  to  which  the   given   equations  reduce  when  we 

Through  these  linear  equations  m^  of  the  differences 
Xy — ax ,  X2, — iZj'  •  •  •  •  ^m — ci^  may  be  expressed  in  terms  of  the  n — m 
remaining,  if  the  determinants  of  the  m>^  order  which  may  be 
formed  out  of  the  m  rows  of  the  c's  are  not  all  zero. 

If,  say,  Ci.i, ,c 


we  have 


o. 


XI 


of  Functions  of  Several  Variables. 


21 


By  means  of  these  equations  we  ma}^  represent  x^ — a^,  x-i, — a-i,., 
....  ;r„, — a„,  as  power-series  in  the  remaining  n — m  differences, 
the  formal  procedure  being  as  follows* : 

We  write  V  =o,  ....  V  =  o,  and  thus  obtain  for  x-^ — a-^, 
....  x^ — «„  expressions  which  represent  the  first  approximations. 
These  are  substituted  in  V  , V  and  the  resulting  ex- 
pressions are  reduced  so  as  to  contain  only  terms  of  the  second 
dimension.  Continuing  this  process,  we  may  represent  the  re- 
quired expressions  to  any  degree  of  exactness  desired. 

We  accomplish  the  same  in  the  following  manner  :  We  write 
for  m  of  the  quantities  x^ — Uxy-.-.x^ — a„  power-series  with  in- 
determinate coefficients;  and  it  is  seen  that  these  coefficients  may 
be  uniquely  determined. 

It  is  a  fundamental  theorem  in  the  theory  of  functions  that 
these  power-series  are  convergent  as  soon  as  the  differences  x — a 
which  enter  into  them  do  not  exceed  certain  limits,  and  further, 
that  these  power-series  satisfy  the  given  equations.  (See  Chap. 
IV,  Art.  25  et  seq.) 

15.  The  problem  of  the  preceding  article  may  be  solved  in 
the  following  more  symmetric  manner,  in  which  none  of  the  vari- 
ables is  given  preference  over  the  others. 

Besides  the  equations  given  above,  we  introduce  others  which 
are  likewise;  expressed  in  power-series: 

Let     c:m+i.i(.^i— «i)+ +c„+i,„  (;»;„—«„) 


Y    =A, 

J\.  m+l 


The  quantities  c  are  arbitrarily  chosen,  in  such    a  manner, 
however,  that  the  determinant. 


'1.1 


-2,1 


,   c 


>     ^2, 


1.2     1 


2     > 


^m ,  1    )     ''  m  2     >      • 
^m+l,li     ''m  +  l,2>      • 

^n.l     »     ^n,2    >      • 


^l,m     I  ''I, m+l     ) 

^2,m     )  ''2.  m+l     > 

^m ,  m     »  '-m.m  +  l    > 

^m+l,m)  ^m  +  l,m+l> 

^ii,m     >  ^11,  mil     ) 


*  See  also  Chap.  Ill,  Arts.  3-6 ;  Chap.  IV,  Arts.  25  and  26, 


''l.n 
^2,11 


■-m  +  l, 11 


o. 


22  Theory  of  Maxima  and  Minima 

Proceeding  in  a  similar  manner  as  in  Art.  14,  we  write  the 

quantities  V  equal  to  zero,   and   we  thus  have  a  system  of    n 

linear  equations  through  which  we  can  express  the  n  differences 
x^—a^, Xr,—a^  through  t^,  t^, /„_„  : 

Xv—qv  =  ey^^  /l  +  e^.2  ^2+  •  •  •  •  +ev.„-m  4-m  + Y    • 
(v=l,2 n). 

With  the   help  of    these   equations  we    can  express  x^ — «!,.... 
^n—cin  as  power-series  in  A.-  ■  •  •  K-m- 

To  do  this  we  again  write    V    =  o,  and  have  only  terms  of 

the  first  dimension.     We  write  the  first  approximations  that  have 

been  thus  obtained  in    V      and  by    retaining   the  terms  of   the 

second  dimension  derive  the  second  approximations,  etc. 

It  may  be  proved  that  these  power-series  in  t  formally 
satisfy  the  given  equation;  that  they  possess  a  certain  com- 
mon region  of  convergence  if  we  give  certain  fixed  limits  to 
I  ^1  I  ,  I  i^2 1  >  •  •  •  •  I  4-m  I  >  that  they  consequently  in  reality  satisfy 
the  equations;  and  finally,  that  all  the  systems  of  values  {^x^,  x-^., 
. . .  .x„)  which  lie  in  the  neighborhood  of  («!,  a2, .  . .  .  a„)  and  which 
satisfy  the  proposed  equations  are  had  in  this  way. 

In  the  theory  of  maxima  and  minima  we  shall  give  strenuous 
proofs  of  the  statements  just  made,  and  in  this  connection  a  the- 
orem will  be  proved  which  is  very  important  in  the  theor)^  of  the 
Calculus  of  Variations.     (See  Chap.  IV,  Art.  25  el  seg.) 

16.  We  make  the  following  application  of  the  theorem  given 
in  Arts.  14  and  15.  In  accordance  with  this  theorem,  if  between 
n  quantities  x^,  X2, .  . .  .x„  there  exist  -m  equations  in  the  form  of 
power-series,  then  the  differences  Xi — ai, ....  x„, — a,„  may  be  ex- 
pressed through  power-series  of  the  n — m  remaining  variables. 
Weierstrass  said:  ''''  Through  the  m  equations  a  structure  of  the 
{n — ni)"'  kind  in  the  realm  of  the  n  quantities  Xi,  X2, . . .  .x^  is 
defined." 

In  virtue  of  the  theorem  of  Art.  15  these  structures  may  be 
expressed  in  manifold  other  ways. 

If  we  introduce  here,  as  in  Art.  15^  the  quantities  ^j,  ^2  •  •  •  •  4-m^ 
we  have  a  symmetric  representation  of  this  structure,  namely: 


of  Functions  of  Several  Variables. 


23 


xv—av=Pv{fi,  t^, 4-m)   (for  v=^l,  2, n). 

It  follows  from  the  method  of  the  derivation  of  these  expressions 
that  they  satisfy  the  given  equations. 

If  we  wish  to  find  all  systems  of  values  which  satisfy  the 


given   equations  and   in  which    |  x^ — a^  \ 


x„ 


-«„ 


I    are  less 


than  p,  we  may  always  assume  p  so  small  that  the  required  sys- 
tems are  represented  through  the  above  formula. 

A  structure  of  the  (« — mf^  kind  is  thus  defined  in  the  realm 
of  the  n  quantities  ,ri,  Xi^....  x„. 

This  theorem  would  be  of  little  importance  if  the  following 
was  not  true:  the  structure  is  not  a  closed  one  within  itself, 
but  a  structure  which  may  be  continued  over  its  boundaries; 
it  is,  as  Weierstrass  expressed  it,  only  an  element  of  a  complete 
structure.  The  question  arises,  how  are  all  the  remaining  ele- 
ments derived  from  this  one.  We  shall  direct  our  attention  in 
the  following  paragraphs  to  the  discussion  of  this  question. 

17.  Suppose,  both  for  simplicity  and  clearness,  that  n — w  =  l; 
then,  if  we  agree  to  represent  x-^,  X2,....x„  as  power-series  of  the 
last  variable  x„,  the  functions  P(x,^ — £?„)  which  have  been  so 
defined  may  be  continued  as  in  Art.  13.  We  need  only  assume  a 
position  a'n  in  the  region  of  convergence  P(Xa — a^)  and  trans- 
form this  series  into  Pi{x„ — a'„).  We  thus  have  in  general  a  con- 
tinuation as  soon  as  the  realm  of  convergence  of  the  second  series 
extends  outside  of  the  original  realm. 

If  we  make  use  of  the  quantity  t,  the  continuation  may  be 
expressed  in  a  much  more  general  and  symmetric  form.  As  this 
quantity  /  is  in  general  arbitrary,  it  is  possible  to  express  Xi,  X2, 
. . .  .x„  through  /  in  many  ways.  Under  the  assumption  that 
n — m=l,  we  shall  limit  this  investigation  to  a  structure  of  the 
form, 

Xv—av=Pv(l)    (v=l,2,....n). 

We  are  thus  freed  from  an  assumption  tacitly  made  that  the 
series  Pv  (/)  begin  with  terms  of  the  first  dimension;  for  we  can 
choose  Pv  (t)  quite  arbitrarily,  the  only  condition  being  that  it 
must  vanish  for  l=o. 

Let  us  take  instead  of  the  variable  t  the  auxiliary  variable  t. 
There  clearly  exists  an  equation  between  /  and  r,  since,  if  in  the 


24  Theory  of  Maxima  and  Minima 

formula  Pi^x^ — a-^,  x^ — aj, .  . .  .x^ — «„),  through  which  t  is  defined, 
we  write 

then  we  have  r  expressed  as  P{t).     Reciprocally  we  may  repre- 
sent ^  as  a  power-series  in  t. 

Suppose  that  t=^a^  T-f-a2T^+  ....,  where  the  a's  must  sat- 
isfy the  condition   that  the   series   converge   for  certain  values 

of  T. 

We  recognize  from  this  that  the  same  element  which  is  rep- 
resented through 

xv — av=Pv  {t) 

may  also  be  represented  through 

Xv  —  av=Pv{r)     (j/=l,  2, .  .  .  .  w). 

From  these  equations  it  is  clear  that  to  every  value  of  t 
there  belongs  a  value  of  t,  and  we  may  assume  r  so  small  that  the 
corresponding  value  of  t  lies  within  the  region  of  convergence  of 
the  original  series  Pv{t).  We  see  that  both  systems  of  formulae 
represent  the  same  systems  of  values. 

On  the  other  hand,  if  Oj   ,  o,  we  may  express  t  as  P{t): 

If  we  substitute  this  expression  in  the  second  system  of  for- 
mulas Xv — av=Pv{T),  there  must  again  appear  on  the  right-hand 
side  Pv(t).  We  may  now  choose  r  and  /  so  small  that  the  power- 
series,  which  represent  them,  converge;  hence  to  every  value  of 
T  there  corresponds  a  value  t  and  to  every  value  of  t  there  corre- 
sponds a  value  t.  Consequently  the  systems  of  values  {xi,  x^, 
....x^  which  belong  to  pairs  of  corresponding  values  /,  t  are 
identical. 

The  structure  may  be  expressed  through  the  one  or  the  other 
system  of  formulae.  We  shall  say  briefly  that  we  have  trans- 
formed the  expressions  of  the  structure — not  the  structure. 

We  may  accordingly  define  the  continuation  of  such  functions 
as  follows: 

Consider  a  system  of  formulae  having  the  form 

Xv  =  ^v(^t^      (^  =  1,  2, . . .  ./^), 


of  Functions  of  Several  Variables, 


25 


where  the  *'s  are  functions  of  t  which  may  be  expressed  in  the 
form  of  power-series. 

In  this  manner  a  structtire  is  defined,  if  to  t  all  values  are 
given  for  which  *i, . . . .  *„  converge. 

A  structure  is  defined  through 

These  two  structures,  which  in  general  have  nothing  in  com- 
mon, may  have  a  common  position;  they  will  then  agree  for  a 
definite  value  /„  of  t  and  a  definite  value  r^  of  t,  so  that 

*v(4)=*v(to)     {v=\,2....n). 

However,  there  is  nothing  of  especial  interest  in  this.  Yet  it 
may  happen  that  in  the  neighborhood  of  these  positions  the  two 
structures  completely  coincide;  that  is,  there  corresponds  to  every 
value  of  /  in  a  certain  neighborhood  of  4  ^  value  of  r  in  a  cer- 
tain neighborhood  of  Tq,  so  that  we  have  continuously 

*v(^)=*v(t)     (v=1,2, n). 

This  is  expressed  analytically,  if  we  write 

where  s  and  o-  are  two  new  variables. 

In  order  that  the  two  structures  correspond  in  the  neighbor- 
hood of  the  two  positions  under  consideration,  <s>v  and  ♦f  must 
so  correspond  to  each  other  that  ^v  goes  into  ^v,  if  we  write 

and  vice  versa,  *v  must  become  *v  if  we  write  cr  equal  to  a  power- 
series  in  s.  Consequently,  to  every  infinitely  small  value  of  5 
(or  0-),  which  is  smaller  than  p,  where  p  is  a  fixed  limit  taken 
sufficiently  small,  there  must  correspond  an  infinitely  small  value 
of  cr  or  (s),  so  that  we  have  the  equation 

<I>.(/o+5)=:^v(To+<r). 

In  a  similar  manner  as  was  seen  above  in  the  case  of  t  and  t, 
there  is  here  an  equation  between  s  and  cr.  This  equation  must 
be  of  the  form  just  given,  and  we  must  further  have  a,  <o,  if  the 
additional  condition  is  made  that  to  every  infinitely  small  value  of 


26  Theory  of  Maxima  and  Minhna 

<r  there  is  to  correspond  only  one  infinitely  small  value  of  s. 
Reciprocalljs  if  ai  is  different  from  zero,  then  to  every  value  of  cr 
there  is  to  correspond  only  one  value  of  s,  since  in  the  develop- 
ment of  cr  in  a  pov^^er-series  of  s  only  integral  powers  can 
appear.  Accordingly,  if  the  tvi^o  structures  have  a  common 
position  and  besides  are  to  coincide  in  the  neighborhood  of  this 
position,  it  must  be  possible  to  derive  the  expressions  of  these 
structures  from  each  other  in  the  manner  indicated. 

If  this  is  the  case,  then  naturall)^  the  tvi^o  structures  fall 
partly  together.  The  remaining  part  of  the  second  structure  is 
to  be  considered  as  a  continuation  of  the  first  structure. 

That  this  is  conformable  with  our  purpose  is  seen  from  the 
fact  that  each  of  the  structures  is  expressible  through  the  same 
analytic  function;  we  must,  therefore,  regard  as  belonging  in  co^n- 
mon  to  one  another  all  structures  that  are  represented  through 
the  same  analytic  function. 

18.  After  the  preceding  article  we  can  make  clear  the  mean- 
ing of  the  complete  structure  that  was  defined  in  Art.  16,  in  the 
following  manner: 

We  start  from  an  elementary  original  structure  G-^.  We  then 
consider  a  second  element  G-i  which  overlaps  with  G-^  in  the  neigh- 
borhood of  any  position.  Then  G^  is  a  direct  continuation  of  Cj. 
In  the  same  way  let  6^3  be  a  direct  continuation  of  G^ ;  then  G^,  is 
an  indirect  continuation  of  G^,  etc.  The  collectivity  of  the  ele- 
mentary structures  G  form  a  complete  structure  closed  in  itself. 

It  makes  no  difference  from  which  element  we  start.  The 
collectivity  of  the  systems  of  values  (;);,,  x^^-.-.x^)  to  which  we 
come  in  this  manner,  is  a  complete  structure. 

Weierstrass  calls  such  a  complete  structure  in  the  realm  of 
the  quantities  x-^,  x^,. . .  -x^  a  monogenic  structure^''  because  all 
the  elementary  structures  of  the  realm  arise  from  one  and  the 
same  source. 

In  accordance  with  this  a  curve  of  the  second  degree  is  a 
monogenic  structure  of  the  first  kind  in  the  realm  {x,y)\  however, 
the  system  of  two  straight  lines  is  a  non-m'Onogenic  structure. 


*  In  connection  with  the  definitions  of  the  structures  given  here,  which  are  adequate 
for  the  present  discussion,  see  a  paper  by  Prof.  Osgood  (Bull,  of  American  Math.  Soc, 
June,  1898). 


of  Functions  of  Several  Variables. 


27 


If  we  have  a  monogenic  structure  of  the  first  kind  in  the 
realm  of  n  quantities,  and  if  we  seek  all  those  positions  in  which 
one  of  the  variables  x  has  a  definite  value,  then^we  may  define  all 
the  appertaining  systems  of  values  of  the  remaining  variables  as 
a  system  of  functions  of  x. 

If  we  take  a  realm  of  two  variables  x,  y  and  seek  all  the 
positions  for  which  x  takes  a  definite  value  x\  the  corresponding 
values  of  y  are  called  values  of  one  and  the  same  analytic  func- 
tion of  X  for  x=x' . 

In  this  structure  y  is  therefore  a  function  of  x,  and  the  func- 
tion has  for  every  given  value  of  x  as  many  values  as  there  are 
positions  on  which  x  has  that  value. 

19.  We  have  now  to  show  that  the  definition  of  Article  18 
agrees  with  that  given  of  an  analytic  function  in  Arts.  11  and  13, 
agrees  in  the  sense:  that  all  the  systems  of  values  of  y  which 
belong  to  x'  are  also  to  be  had  in  the  manner  given  in  Arts.  11 
and  13,  with  the  distinction  that  here  we  have  still  other  values 
of  y  which  cannot  be  included  in  the  previous  definition. 

A  difficulty  that  attended  the  last  definition  may  be  avoided: 
We  start  from  the  representation  y=P{^x — a)  and  derive  from  it 
all  the  possible  power-series.  We  have  thus  defined  a  function  of 
X.  This  structure  has,  however,  boundary  positions,  and  we  have 
yet  to  determine  whether  the  boundary  valvies  are  to  be  counted 
with  the  structure. 

If  we  make  the  explanation  of  the  structure  as  in  Art.  18, 
then  this  difficulty  is  avoided. 

We  consider  ov\\  those  pairs  of  values  to  which  we  can  in 
reality  come,  as  belonging  to  the  structure  which  is  defined  through 
the  given  function  of  x,  y. 

Suppose  that  we  again  have   the   two  equations 

x=P,{t),y=P^{t) ....(1) 

For  /— 4.  let  x=a,  y=b.     The  developments  are  therefore: 


X- 

y- 


If  wne  choose  a.-^\o,  we  have  from  the  first  of  these  equations 

1r-t^^Pix—a). 


28  Theory  of  Maxima  and  Minima 

Now,  write  this  expression  in  the  second  of  the  equations,  and 
it  follows  that 

:y—d==P{x—a) (2) 

The  power-series  on  the  right-hand  side  vanishes  for  x^a.  If 
we  limit  ourselves  to  the  values  (x,  y)  which  lie  within  a  certain 
neighborhood  of  the  position  {a,  b),  the  positions  which  are  rep- 
resented through  the  equations  (1)  are  also  represented  through 
the  equation  (2). 

For  this  to  be  true: 

(1)  X  must  be  chosen  so  near  a  that  t — 4  may  in  reality  be 
represented  through  a  power-series  in  x — a  ; 

(2)  X  must  lie  so  near  a  that  4  lies  within  the  region  of  con- 
vergence of  the  power-series  in  t  ; 

(3)  If  we  write  the  expression  of  x — a—Pi{t — 4)  in  equa- 
tion (2),  the  resulting  series  must  be  convergent. 

These  conditions  may  always  be  satisfied,  and  we  may  there- 
fore say  that  the  equation  (2)  represents  the  same  system  of 
values  in  the  neighborhood  of  {a,  b)  as  the  two  original  equa- 
tions (1). 

Reciprocally,  if  we  start  with  such  a  power-series  as  (2)  and 
derive  from  it  two  others  (1),  then  the  structure  which  is  thus 
formed,  that  is,  the  collectivity  of  the  systems  of  values  {^x,  y)  is 
one,  whose  definition  is  contained  in  the  one  just  given.  For  by 
writing  x — a^t,  we  have  y=P(t).  We  take  in  the  region  of 
convergence  P{x — a)  the  position  a^  and  transform  P(x — a)  into 
P{x — «i).  If  we  then  write  x — a^=t\  we  have  t'  —  t+a — a^  or 
t=t' -\-ai — a.  The  transformation,  therefore,  consists  in  the  in- 
troduction of  t'  instead  of  /  in  the  expression  ior  y. 

If,  accordingly,  we  derive  from  y=^P{x)  all  the  possible 
power-series  and  write  each  of  them  equal  to  y,  the  structure 
{x,  y)  which  has  been  so  defined  is  certainly  contained  in  a  mono- 
genic structure. 

This  structure,  in  accordance  with  the  definition  of  Art.  13, 

can  contain  yet  certain  other  positions ;  for  if  we  give  to  4  such 

a  value  that  ai=o,  then  t — 4  cannot  be  written  equal  to  ^{x — a). 

X. . 

Moreover,  we  have  t — tQ:i=P{v'x — a),  if  the  power-series  in  / — 4. 

through  which  x — a  is  expressed,  begins  with  (/ — 4)'^.  and  there- 
fore y  is  expressed  in  general  through  a  power-series  of  a  definite 


of  Functions  of  Several  Variables. 


29 


root  of  X — a.  These  are  the  singular  systems  of  values,  the 
boundary  positions,  which  we  must  count  as  belonging  to  the 
structure.  For  this  reason  the  explanation  given  in  Art.  18  of 
the  structure  is  to  be  preferred  to  the  one  in  Arts.  11,  13  and  16, 
since  the  first  affords  the  boundary  positions,  and  besides  shows 
extraordinary  symmetry. 

20.  The  considerations  given  in  the  previous  paragraphs  for 
the  case  of  two  variables  may  be  easily  extended  to  any  number 
of  variables,  say  n.. 

Write 

;i;— a  =  ai(/f— 4)  +  a2(if— 4)2+ , 

y—d=^,  (^— 4)+/32  (/— 4)2+  . . . ., 


where  at  least  one  of  the  quantities  aj,  jSj,  yi, . . . .  is  different  from 
zero,  say  a ;  then  I — 4  ^^J  be  represented  as  a  power-series  in 
X — a,  and  then  likewise  y — d,  z — c, ....  as  power-series  in  x — a. 

If  we  continue  these  power-series  by  means  of  the  same 
intermediary  values,  we  have  a  system  of  n — 1  functions  of  x. 
This  system  is  contained  in  our  structure.  If,  however,  all  the 
quantities  aj,  /Sj, . . .  .  were  equal  to  zero,  then  we  would  have  posi- 
tions which  could  not  be  contained  in  the  definitions  given  in  Art. 
13,  and  which  must,  moreover,  be  still  counted  as  belonging  to 
the  structure. 

We  may  accordingly  define  a  system  of  analytic  functions  of 
one  variable  x  in  the  following  manner : 

Let  any  monogenic  structure  of  the  first  kind  in  the  realm  of 
the  quantities  x-^,  x^,,. . .  .x^  be  given ;  if  we  consider  the  positions 
at  which  x  has  one  and  the  same  value,  then  we  have  either  only 
one  such  position,  and  then  x-^,  X2,....x„  are  one-valued  functions 
of  X,  or  there  are  several  such  values  and  then  x^y  Xj,, . . .  .x„  are 
many -valued  functions  of  x. 

We  may  also  express  this  as  follows:  If  we  consider  x-^,  x^, 
....;»;„  as  functions  of  x^  then  we  have  all  the  systems  of  values 
i^x^,  X2,...  .Xa)  that  belong  to  a  definite  value  of  x,  if  we  keep  in 
sight  all  those  positions  in  which  x  has  the  definite  value  under 
consideration. 


30  Theory  of  Maxima  and  Minima 

21.  At  the  close  of  this  introduction  to  the  General  Theory  of 
Functions  we  must  yet  consider  systems  of  values  of  a  structure 
which  lie  at  infinity. 

Weierstrass  also   made  use  of   the  sign  x — a  for  the  case 

a  =  co  and  understood  by  it  nothing  other  than  — ,  that  is,  a  linear 

X 

function  which  vanishes  for  x=^a=^ .  We  may  therefore  express 
—  through  power-series  as  has  been  done  with  x — a  for  finite 

X 

values  of  a.     Our  definition  is  then  true  in  general. 
When  several  variables  are  present,  we  write 
xv — av=Pv{t)     {v=\,  2,.  ■ .  .n). 
The  power-seaies  on  the  right  vanishes  for  t=o,  and  Xv  goes  into 
av.     If  a  V  is  infinite,  this  means  —  may  be  expressed  through 

Xv 

such  a  power-series.  If,  further,  these  formulae  are  in  reality  true 
{i.  e.,  if  the  series  are  convergent),  then  the  position  (iZi,  a^, .  .  .t?„) 
belongs  to  the  structure.  In  this  manner  an  element  of  the 
structure  is  defined  which  lies  at  infinity. 

A  position  {a^,  a^^.  .  .  .a„)  is  always  at  infinity  if  only  one  of 
the  quantities  a  is  infinite. 


of  Functions  of  Several  Variables. 


31 


chapti:r  ii. 

MAXIMA   AND    MINIMA   OP    FUNCTIONS    OP    SEVERAL   VARIABLES 
THAT  ARE   SUBJECTED   TO   NO   SUBSIDIARY   CONDITIONS. 


1.  Introduction. — Mathematician.s  have  always  been  occu- 
pied with  questions  of  maxima  and  minima.  Already  in  Euclid's 
time  one  of  the  simplest  problems  of  this  character  was:  Find 
th&  shortest  line  which  may  be  dravjn  from  a  point  to  a  line. 
Methods  for  the  solution  of  such  problems  were  known  long 
before  the  differential  calculus,  and  certainly  such  problems  exer- 
cised considerable  influence  upon  the  discovery  of  the  calculus. 
Indeed,  Lagrange  wished  to  consider  Fermat  as  the  discoverer  of 
the  calculus,  as  he  had  been  occupied  with  many  problems  of 
maxima  and  minima.  He  would  in  reality  have  probably  come  to 
the  differential  calculus  had  he  started  from  a  somewhat  more 
general  point  of  view.  The  theory  of  maxima  and  minima  was 
rapidly  developed  along  the  lines  of  the  calculus  after  its  discov- 
ery. One  was  at  first  satisfied  with  finding  the  necessary  condi- 
tions for  the  solution  of  a  problem.  These  conditions,  however, 
are  seldom  at  the  same  time  also  sufficient.  In  order  to  decide 
this  last  point  the  discovery  of  further  algebraic  means  was 
necessary. 

2.  We  shall  presuppose  in  the  following  discussion,  unless  it 
is  expressly  stated  to  the  contrary,  that  not  only  the  quantities 
that  appear  as  arguments  of  the  functions,  but  also  the  functions 
themselves  are  real,  and  that  the  functions,  as  soon  as  the  vari- 
ables are  limited  to  a  definite  continuous  region,  have  within  this 
region  everywhere  the  character  of  one-valued  regular  functions. 
We  define  regular  functions  in  the  following  manner:  A  func- 
tion f{x)  is  regular,  if  the  function  is  defined  for  all  values 


32  Theory  of  Maxima  and  Minima 

of  X  within  certain  limits  of  x,  and  if  further  for  any  value  a 
of  the  argument  within  these  lim,its,  the  development 

f{a  +  h)=f{a)  +  hf'{a)+  ^f"{a)+  .... 

is  possible ;  the  series  must  be  convergent  and  m^ust  in  reality* 
represent  the  values  of  the  function. 

If  we  have  a  function  of  several  variables  f{x-^,  x^,...  .x^ 
and  if  a-^,  a^,. . .  .a^  are  any  values  of  the  variables,  then  the  func- 
tion is  said  to  be  regular,  if  the  development 

f{a^  +  hi,  a:,+  h^, . . .  .a^-{-  h„)=P{h^,  h:„....h„) 

is  possible;  every  term  of  the  power-series  is  here  a  product 
of  a  power  of  h  and  a  factor  that  is  independent  of  h. 

We  limit  ourselves  to  such  functions  that  are  analytic  struc- 
tures of  the  nature  described  in  Arts.  8-21  of  the  previous  chap- 
ter; only  for  such  functions  can  we  derive  general  theorems,  since 
for  other  functions  even  the  rules  of  the  differential  calculus  are 
not  applicable. 

The  problem  of  finding  those  values  of  the  argument  of  a 
function /"(;»;),  for  which  the  function  has  a  maximum  or  minimum 
valijie  is  not  susceptible  of  a  general  solution,  since  there  are 
functions  which,  in  spite  of  the  fact  that  they  may  be  defined 
through  a  simple  series  or  other  algebraic  expressions  and  which 
vary  in  a  continuous  manner,  have  an  infinite  number  of  maxima 
and  minima  within  an  interval  which  may  be  taken  as  small  as  we 
wish.    Such  functions  do  not  come  under  the  present  investigation. 

3.  We  say  t  that  a  function  f{x)  of  one  variable  has  at  a 
definite  position  x—a,  a  m^aximum  or  a  minimum,  if  the  value 
of  the  function  for  x=^a  is  respectively  greater  or  less  than  it 
is  for  all  other  values  of  x  which  are  situated  in  a  neighbor- 
hood I X — a  I  •<  8  «5  near  as  we  wish  to  a. 

The  analytical  condition  that  f{x)  shall  have  for  the  position 
x=^a, 

a  maximum,  is  expressed  hj  f{x) — f{d)<io  ;  )     r     i    i      » 

a  minimum,  is  expressed  ^j  fix) — f{a)  >o  ;  )  ' 

In  the  same  way  we  say  a  function  f{xi,  x^,. .  .x„)  of  n  variables 


*  And  not  only  formally.     (See  Art.  IS.) 

t  See  Annals  of  Mathematics,  Vol.  IX,  No.  6,  p.  187. 


of  Functions  of  Several  Variables. 


33 


has  at  a  definite  position  Xi^a^,  X2=a2, .  . . . x„=a„  a  maxiimiin 
or  a  minimum,  if  the  value  of  the  function  for  Xi=^ai,  X2=a2, 
. . .  .Xa=aa  is  respectively  greater  or  less  than  it  is  for  all  other 
systems  of  values  situated  in  a  neighborhood 

\x\—ax\<^x    (X=l,  2, n) 

as  near  as  we  wish  to  the  first  position  ;  and  the  analytical  con- 
dition that  the  function /'(;i:i,  X2, . . .  .Xa)  shall  have  at  the  position 
Xi=^ai,  X2^=^a2t  •  ■  •  .^„=iz„ 

a  maximum,  is  '.fixi,  X2, . . .  .x^^fi^a^,  aj-  •  •  •  -^n)  <<?. 
a  m.inim^um,  is  :f(xi,  X2, . . .  .x„)—f(ai,  a2, . . .  .a„)y-o, 

for  |;i;x— «xi  <Sx     (X=:l,  2, n). 

4.  The  problem  which  we  have  to  consider  in  the  theory  of 
maxima  and  minima  is,  then,  to  find  those  positions  at  which  a 
maximum  or  minimum  really  enters. 

Since  this  problem  for  functions  of  one  variable  is  exhaust- 
ively treated  in  the  elements  of  the  differential  calculus,  we  shall 
discuss  it  here  very  briefly. 

If  x^,  X2  are  two  values  of  x  situated  sufficiently  near  each 
other  within  a  given  region,  then  the  difference  of  the  correspond- 
ing values  of  the  function  is  expressible  in  the  form : 

f(x2)-f(x,)^^,  (;i:x  +  €(;r-;t;,)), 

X2      Xi 

where  c  denotes  a  quantity  situated  between  o  and  1 ;  or,  if  Xi  is 
written  =x  and  X2  =x+h  : 

[1]  f(x+h)-f(x)  =  hf'  (x  +  e  h). 

From  this  theorem  may  be  derived  Taylor's  theorem  in  the 
form,* 

[2]     f(x+h)-f(x)^hf'(x)  +  j^  hY"  (x)+.... 

+  7-^  h"-'f'"'^  (x)  +  -\  h"f"^  (x  +  e  h). 
\n — l)\  n\ 

In  the  two  formulae  last  written,  instead  of  x-\-h,  write  x 
and  write  a  in  the  place  of  x ;  they  then  become 


*See  Jordan,  Cours  D' Analyse,  T.  I,  \l  249-250. 


34  Theory  of  Maxima  mid  Minima 

and 

[2-]    /(.T)-/(a)  =  ^/-(«)+(£zplV"{«)  +  .... 

Since  /"(a;)  is  a  regular,  and  consequently  continuous,  func- 
tion, the  same  is  true  of  all  its  derivatives.  If  /'  (a)  is  different 
from  zero,  then  with  small  values  of  /i=x — a,  the  value  of 
/"(«  +  €  k)  is  different  from  zero  and  has  the  same  sign  as  /'  (a). 

According  to  the  choice  of  /i,  which  is  arbitrary,  the  differ- 
ence/'(;i:)-^  (a)  can  be  made  to  have  one  sign  or  the  opposite 
sign,  if  /'  (a)  is  either  a  positive  quantity,  or  if  it  is  a  negative 
quantity.  Hence  the  function  /{x)  can  have  neither  a  maximum 
nor  a  minimum  value  at  the  position  x^a,  if  y  (a)  ^o. 

We  therefore  have  the  theorem:  Maxima  and  minir/ia  of 
the  function  f{x)  can  only  enter  for  those  values  of  x  for 
which  f  ix)  vanishes. 

It  may  happen  that  for  the  roots  of  the  equation/"'  {x)=o, 
some  of  the  following  derivatives  also  vanish.  If  the  /?"'  deriva- 
tive is  the  first  one  that  does  not  vanish  for  the  root  x^a,  then 
from  equation  [2"^]  we  have  the  formula: 

/(;*;)-/(«)=  i^I^V"'  {x^^{x~a)), 
n  \ 

and  with  small  values  of  h^x — a,  owing  to  the  continuity  of 
_/"<"'  {x^  the  qriantity  /"'"'  (a  +  e  h)  will  likewise  be  different . from 
zero  and  will  have  the  same  sign  as  /"'"'  (a).  If,  therefore,  n  is 
an  odd  integer,  we  may  always  bring  it  about,  according  as  h  is 
taken  positive  or  negative,  that  the  difference  f{x) — f{a)  with 
every  value  of  /"'"'  {a)  has  either  one  sign  or  the  opposite  sign; 
consequently  the  function  f{x)  will  have  at  the  position  x~-^a, 
neither  a  maximum  nor  a  minimum  value. 

If,  however,  n  is  an  even  integer,  then  A"  is  always  positive, 
v/hatever  the  choice  of  h  may  have  been;  consequently  the  differ- 
ence/"(^) — f{<^)  is  positive  or  negative  according  as  f^""^  (a)  is 
positive  or  negative. 

In  the  first  case  the  function /"(;!;)  has  a  minimum  value  at 
the  position  x=a,  in  the  latter  case  a  maximum. 


of  Functions  of  Several  Variables. 


35 


Taking  this  into  consideration,  we  have  the  following  theo- 
rem for  functions  of  one  variable: 

Maxima  and  minima  of  the  function  f  i^x)  can  only  enter 
for  the  roots  of  the  equation  f  {^x^=o. 

If  a  is  a  root  of  this  equation,  then  at  the  position  x=^a 
there  is  neither  a  maximum  nor  a  minimum.,  if  the  first  of 
the  derivatives  that  does  not  vanish  for  this  value  is  of  an  odd 
degree;  if,  however,  the  degree  is  even,  then  the  function  has  a 
m^aximum  value  for  the  position  x=^a,  if  the  derivative  for 
x=^a  is  negative;  a  minimum,  if  it  is  positive. 

5.  In  order  to  derive  the  analog  for  functions  of  several  vari- 
ables, v/e  start  again  with  Ta}4or's  theorem  for  such  functions. 
We  may  derive  this  theorem  by  first  writing  in  /^(^i,  X2,...  .x^) 

xx=ax  +  u{x)c-a)0,     (^=1,  2, n\ 

where  «  is  a  quantity  that  varies  between  o  and  1;  we  then  apply 
to  the  function 

^  {u)==^f{ai^u(Xi — <^i),  a^  +  uix^ — (22)1-  •  •  .aa  +  u{x„ — «„)) 

Maclaurin's  theorem  for  functions  of  one  variable,  viz.: 


[3] 


u 


W 


i>{u)=cf>(o)+  -fpf  (C7)+  -^<l>"{o)+  .... 
(»^ — Ijl  m^l 


and  in  this  expression  write  u=l,  as  follows: 

For  brevity,  denote  by  f^  (x^,  Xj,....  x^)  the  first  derivative  of 

f(xi,  X2, x„)  with  respect  to  x^,  hy  fi,^^^^(xi,  x^, x^  the 

derivative  of  f{xi,  X2,...  .x„)  with  respect  to  ^ttiand  x^,^,  i.  e., 

/r  /„  \       d^  f  (^Xi,  X2,  . . . .  Xr)       , 

ox^^^dx^^ 
We  have,  then, 


k=n 


^'  (^)=2  V ''^^'+^^^i' 


lt-=l 


^), a^  +  u{x,,—a„))(x^—a^V,, 


36  TTieory  of  Maxima  and  Minima 


ki,k2,...k„_i 


ki,k2,...kni 

Hence,  from  [3]  we  have 

k 
+  ' 


*^1  J  *^2»  "•  **  itt— 1 


,^^  (/kj,fcj,  ...u„(«i  +  e  «^(^i— «i), .  ...  ) 

w!  -*^  I  a^  +  €U(x„—a„))(Xi,^—a^^) . ..  (^k„— ^„)) 


k  i,k3,...kn. 

From  this  it  follows,  if  we  write  ^=1: 


k 


'i.'^a 


+ 


I  1  X://k,>k,...k^_^(gl,g3.--'gn)   (^k— «ki)----  ) 


ki,k2...k^_l 


1_^  (/ki,k2....k„(^i  +  e(^i— ^i),  a^  +  eixi—a^),....  \ 


kl,k2,--Km 


of  Functions  of  Several  Variable:. 


37 


6.  We  are  not  accustomed  to  Taylor's  theorem  in  the  form 
just  given;  in  order^  to  derive  this  theorem  as  it  is  usually  given, 
we  observe  that  upon  performing  the  indicated  summations,  each 
of  the  indices  /J^i,  k-^,.  .  ■  ■■,  independently  the  one  from  the  other, 
takes  all  values  from  1  to  n,  so  that  the  X'"  term  in  the  develop- 
ment is  a  homogeneous  function  of  the  X^''  degree  in  x-^ — a^,  x-^ — a^, 
....  x^ — «„.  The  general  term  of  this  homogeneous  function  may 
be  vi^ritten  in  the  form 

~  D.N.{x^—a^  \xi—a^--{x^—a^  ", 

where  Xj  +  Xj-f  ....  -f  X„=X, 

'd^  f{x^,XT,,...xy 


(a   f  yXx,  Xii .  .  . ^„ ) \ 
-tI; — ;n; ->  x    I 


On 


_     -(A1  +  A2+ +  A.„) 

and  N  is  the  number  of  permutations  of  n  elements  of  which 
Xj,  X2, . . . .  X„  respectively  are  alike, 

^•^--^=X.!X2!....X„!- 
Further,  writing  Xi, — «k  =  ^k>  we  have  finally: 
[4]    /(xi,  X:i,...x„)—/{ai,  a2,...a„) 


+ 


+ 


n. —  u 

2Ud/(xi,  X2, ;i;,.)\ 
U             dx^  J 

k=l 

2^\\  dx^x^         ) 


h^ 


a\ ,  a2, (In 

ai  ,a2  , a  n 


\l+\2- 


+2  {f'"'' 


Xl,X 


l.'^2. 


Xl+X2  + 


.X„ 


X,!  X^L.-XJ. 


Xl,  X2, x„ 


X,!  \^\...\S 


38  Theory  of  Maxima  and  Minima 

This  is  the  usual  form  of  Taylor's  theorem  for  functions  of 
several  variables.  In  particular,  vv^hen  ^w=l,  the  above  develop- 
ment is :  , 

[5]       f{x^,   Xi_ ^„)— /  («!,    «2. «n) 

k=n 


k=l 


The  function  f  {x-^,. . .  .x„)  is  regular  and  continuous,  as  are 
consequently  all  its  derivatives.  If,  therefore,  the  first  deriva- 
tives oi /{xi,  X2,. . .  .x„)  a.re  all,  or  in  part,  <o  for  x^=ai,. . . . 
Xn  =  ci„,  then  they  vi^ill  also  be  different  from  zero  for  x^^ai  +  e  ki, 
.  . .  .x„=a„-\-e  k„,  where  the  absolute  values  of  Ai,  A2, . . . .  h„  have 
been  taken  sufficiently  small ;  these  derivatives  w^ill  also  be  of  the 
same  sign  as  they  were  for  Xi=ai,  Xi—a^, . .  ■  .x„=^a„.  If  now  we 
choose  all  the  A's  zero,  with  the  exception  of  one,  which  may  be 
taken  either  positive  or  negative,  we  see  that  when  the  corre- 
sponding derivative  has  one  sign  or  the  opposite  sign,  we  may 
always  bring  it  about  at  pleasure  that  the  difference 

f(xi,X2, ;r„)— /(«!,  «2, a„) 

is  either  a  positive  or  a  negative  quantity,  and  consequently  at 
the  position  ai,  a2, . . .  .a-n  neither  a  maximum  nor  a  minimum  value 
of  the  function  is  permissible. 

We  therefore  have  the  following  theorem: 

Maxima  and  minima  of  the  function  f  (xi ,  X2,. . .  ■  x„)  can 
only  enter  for  those  systems  of  values  of  {x^,  X2,  ■ . .  .x^  which 
at  the  sam^e  time  satisfy  the  n  equations: 

[5]  |/.o,     |/.. Y-^o. 

a  Xi  0X2  ox„ 

It  may  happen  that  for  the  common  roots  of  the  system  of 
equations  [6]  also  still  higher  derivatives  vanish.  In  this  case  we 
can  in  general  only  say  that,  if  for  a  system  of  roots  of  the 
equations  [6]  all  the  derivatives  of  several  of  the  next  higher 
orders  vanish,  and  if  the  first  derivative  which  does  not  vanish 
for  these  values  is  of  an  odd  order,  the  function,  as  may  be  shown 
by  a  similar  method  of  reasoning  as  above,  has  certainly  no  maxi- 
mum or  minimum  value. 


of  Functions  of  Several  Variables. 


39 


7.  If,  however,  this  derivative  is  of  an  even  order,  then  in 
the  present  state  of  the  theory  of  forms  of  the  «"*  order  in  sev- 
eral variables  there  is  no  general  criterion  regarding  the  behavior 
of  the  function  at  the  position  in  question.  We  therefore  limit 
ourselves  to  the  case  where  the  derivatives  of  the  second  order 

of  the  function  f  {,Xy,  x^, x^)  do  not  all  vanish  for  the  system 

of  real  roots  a^,  a^, . . .  .a„  of  the  equations  [6]. 

In  this  case  we  have  a  criterion  in  the  formula, 

[7]    /(^i,  ^2,....;r„)— /(»!,  ai,....a„) 

2  -'^   y\  dx^dx  lai  +  eAu an+€/ia) 

by  which  we  may  determine  whether /'(;«;i,  X:^, . . .  .x„)  has  a  maxi- 
mum or  a  minimum  value  on  the  position  a^  ^2, . . . .  a,„  since  we 
may  determine  whether  the  integral  homogeneous  function  of  the 
second  degree. 


f^[\  dx^dx^  'ai  +  .Ai, an 


}. 


in  the  n  variables  ^„  Aj, . . . .  A„  is  for  arbitrary  values  of  those 
variables  continuously  positive  or  continuously  negative. 

On  account  of  their  presupposed  continuit}^  the  quantities 


I^VJXi,  X2, Xn)\  and  /' 

\  d  X^    d  X^  jai+i/li, an'€/ln\ 


d  J  \Xx,  Xj, ....  x„) 

9^x   ^^^ 


•) 

I  ai, an 


with  values  of  A„  A2, . . . .  A„  taken  sufficiently  small,  differ  from 
each  other  as  little  as  we  wish  and  are  of  the  same  sign;  hence, 
with  small  values  of  the  ^'s  the  functions 


\n 


a^/(^,_^^„)i^^  A^ 


^  X\   3^/i  I  a\  +  fhi an  + 

^  \\  dx-^dx  I  ax, a„      ) 


fhn     ^ 


\l>- 


have  always  the  same  sign,  and  we  may  therefore  confine  our- 
selves to  the  investigation  of  the  latter  function. 


40  Theory  of  Maxima  and  Minima 

If  we  find  that  through  a  suitable  choice  of  Ai,  h^,....h^  the 
expression     ' 

can  be  made  at  pleasure  either  positive  or  negative,  the  same  will 
be  the  case  with  the  difference  /{x^,  x^,. . .  .x„)—/'{ai,  a^.,. . .  .«n)» 
and  consequently  f {^x^,  Xi,....x^  has  on  the  position  (ai,  a^., 
. . .  .a„)  neither  a  maximum  nor  a  minimum  value. 

We  therefore  have  as  a  second  condition  for  the  existence  of 
a  maximum  or  a  minimum  of  the  function  XC^n  -^21  •  •  •  -^n)  on  the 
position  («!,  i?2>-  •  •  -^n)  that,  in  case  the  second  derivatives  of  the 
function /"(j^i,  ;i:2, .  . .  .  ;i^„)  do  not  all  vanish  at  this  position,  the 
homogeneous  quadratic  form 

f^\\  dx^dx  Iai,a2, an     ' 

must  be  always  negative  or  always  positive  for  arbitrary  values 
of  ^1,  /^2, /?„. 


THEORY  OF  THE  HOMOGENEOUS  QUADRATIC  FORMS. 

8.  There  are  two  kinds  of  integral  homogeneous  functions  of 
the  second  degree,  or,  as  they  are  usually  called,  quadratic  forms, 
viz.: 

I.,  formce  indefinita;'*  which  with  real  values  of  the  vari- 
ables can  become  both  positive  and  negative,  and  that,  too,  for 
values  of  the  variables,  whose  absolute  values  do  not  exceed  an 
arbitrarily  small  quantity; 

II.,  formce  definitce,  which  with  real  values  of  the  variables 
have  always  the  same  sign. 

We  distinguish  among  the  definite  forms: 

[1]  Those  which  only  vanish,  when  all  the  variables  become 
zero,  and 


'^ Gauss,  Disq.  Aiithm.,  p.  271. 


of  Functions  of  Several  Variables. 


41 


[2]  Those  which  may  also  vanish  for  other  values  of  the 
variables.  • 

If  our  homogeneous  function  is  an  indefinite  form,  it  is  clear 

that  the  function /(;ri,  x^, x^  has  neither  a  maximum  nor  a 

minimum  upon  the  position  (^i,  a^,...  .«„);  for  if  the  right-hand 
member  of  [7]  is  positive  (say)  for  a  definite  system  of  values 
of  the  ^'s,  then  in  accordance  vi^ith  the  definition  of  the  indefinite 
quadratic  forms  we  can  find  in  the  immediate  neighborhood  of 
the  first  system  a  second  system  of  values  of  the  ^'s,  for  which 
the  right-hand  side  of  the  equation  [7]  is  negative;  consequently, 
also,  the  difference 


/(^l-  ^2 ^n)— /(«1.  «2 


«„) 


is  negative,  so  that  therefore  no  maximum  or  minimum  is  permis- 
sible for  the  position  (^i,  ofj, . . . .  a„). 

If,  then,  the  second  derivatives  of  the  function /"(;ri,  x^,.  ■  .  x„) 
do  not  all  vanish  at  the  position  (ai,  (Zj,  •  ■  •  ■««)>  it  follows  besides 
the  equations  [6]  as  a  further  condition  for  the  existence  of  a 
maximum  or  a  minimum  of  the  function /"(;tri,  X2,...  .x-^  that  the 
terms  of  the  second  dimension  in  [4]  must  be  a  definite  quadratic 
form,  and,  indeed,  as  will  be  show^n  later,  one  which  can  only 
vanish  when  all  the  variables  become  zero. 

The  question  next  arises:  Under  what  conditions  is  in  gen- 
eral a  homogeneous  quadratic  form 

[8]  <i>{x^,X2, x„)  =  ^Ax^^Xxx^ 

a  definite  quadratic  form  of    the  kind  indicated  ? 

'  9.  Before  we  endeavor  to  answer  this  question,  we  must  yet 
consider  some  known  properties  of  the  homogeneous  functions  of 
the  second  degree. 

Suppose  that  in  the  function  ^  {^x-^,  x^,. . .  .x„)  in  the  place  of 
{xi,  X2, . . . .  Xa)  homogeneous  linear  functions  of   these  quantities 

[9]  :yx-^^cx^i,x^      {\=l,2,...,n) 


42 


Theory  of  Maxima  and  Minima 


are  substituted,  which  are  subjected  to  the  condition  that  in- 
versely the  x'^  may  be  linearjy  e;i;pressed  in  terms  of  the  7's,  and 
consequently  the  determinant 


[10] 


^11    >     ^12)     • t^ln 

^21    '     ^22    )      ^211 


^nl   t     ^n2  > 


^1   ^  ^11  ^22- 


n  ii< 


o. 


The  function  ^  (;iri,  ;i;2, x„)  is  thereby  transformed  into 

[11]  <^(^l.   ^2. ^n)=«/'(J^'l.>'2.••••:^^n). 

Owing  to  this  substitution  it  may  happen  that  i/»  (jFi,  ;V2, 
does  not  contain  one  of  the  variables  y,  so  that  <^  (;Vi,  x^,. 
is  expressible  as  a  function  of  less  than  n  variables. 
In  order  to  find  the  condition  for  this,  let  us  write 


[12]  ,^;,  =  |^|i_  =  2^x/.^/.      (X=l,2....^). 

If  i/»  is  independent  of  one  of  the  y s,  say  y^ ,  so  that  conse- 


quently 


^  y^ 


=0,  then  from  the  n  equations 


i>.=n 


li==n 


M         2  ^.=2(1^  l^j-Sf^.x^) 

(A  =  l,  2,....«) 
we    may   eliminate    the   m — 1    unknown    quantities    ^    ,     *^    , 


'9>'n-l 


We  thus  have  among  the  ^'s  an  equation  of  the  form 


[14] 


2'^'»*^' 


.=0, 


where  the  ^'s  are  constants. 

Owing  to  equations  [12]  this  means  that  the  determinant  of 
the  given  quadratic  form  vanishes,  i.  e., 


[15] 


^1  =t  -^11   -^22  •  •  •  •■^x, 


—  O. 


of  Functions  of  Several  Variables. 


43 


Reciprocally,  it  is  easy  to  show  that  if  the  equation  [15]  is 
true,  it  is  possible  to  express  the  function  <^  as  a  function  of  n — 1 
variables. 

For  we  have 


[16] 


X  X, /t 

2l  *^x(^'i.  X^,...X^)  ^x|  =  2  ^X/x^V^A-' 

X  X, /i 

and  consequently 

[17]     2  { ^^  (^1'  ^2. ^n)  ^'\|  =  2  { *^^  (^'i.^'a.  •  • .  •  ^'«  )f  \ |- 

There  exists,  further,  the  well-known  Kuler's  theorem  for  homo- 
geneous functions: 


[18] 


2{'^^(^l'   ^2. ^n)^x|  =  «^(^l,  ^2, ^n)- 


If  now  we  assume  that  equation  [15],  or,  what  amounts  to 
the  same  thing,  an  identical  relation  of  the  form  [14J  exists,  and 
if  we  substitute  in  (f>  (xi^Xj,.  .x„)  the  quantities  ;t^x+/ X:x  in  the 
place  of  xx  (X=:l,  2,....n)  and  develop  with  respect  to  powers 
of  /,  we  then  have 

<i>{Xi  +  i  ^l,Xi+t  i;2^ ^n  +  f  ^n)=<j>(Xi,X2, X„) 

+  2   t   2{^X<^x(^1.^2.----^n)|+^^<^(>^l''^2----'^n). 

It  follows,  when  we  take  into  consideration  the  equations 
[14]  and  [18] ,  since  the  equation  [14]  is  true  for  every  system 
of  values  (x^,  x^,...  -x^),  that 

<^  {x^  +  t  k^, ;c„+ 1  k^)=<l>  {x^, x^). 

Hence,  if  the  equation  [15]  exists,  or  if  the  ^'s  satisfy  the 
equation  [14]  for  every  system  of  values  {x^,  X2,....x„),  then 
<j)  {xi,  Xi,. . .  .x„)  remains  invariantive,  if  x^+  t  ky^  is  written  for 
x-^,  where  t  is  an  arbitrary  quantity. 


44  Theory  of  Maxima  and  Minima 

Consequently,  it  being  presupposed  that  ^^.^o,  if  /  is  so  de- 
termined that  the  argument  Xv^t  kv^o,  we  have 

[19]         <^{x^,X:^, X„)=<l>[x,—  ^Xv,X^---^Xy, 

.  .  .  .,Xv~\ ——-Xv,  O,  Xy+i —  ~^X„,.  .  .  .Xn -,-  Xp], 

Kv  kv  kv      I 

where  4>  is  expressed  as  a  function  of  less  than  n  variables.     We 
therefore  have  the  theorem: 

The  vanishing  of  the  determinant  ^  --^n  ^22-  •  •  --^nn  is 
the  necessary  and  sufficient  condition  that  a  homogeneous 
quadratic  function  (ft  (x^,  x^, x^J^^A^^  x^  x^  be  express- 

ible  as  a  function  of  n — 1  variables. 

10.  We  return  to  the  question  proposed  at  the  end  of  Art.  8; 
and  in  order  to  have  a  definite  case  before  us,  we  shall  assume 
that  the  problem  is:  Determine  the  condition  under  which  the 
function  (j>  (x^,  x^,. . .  .x„)  is  continuously  positive.  The  second 
case  where  ^  {x^,  x^,.  .  .  .x^)  is  to  be  continuously  negative  is  had 
at  once,  if  we  write  — <^  in  the  place  of  (j). 

We  shall  first  show,  following  a  method  due  to  Weierstrass,* 
that  every  homogeneous  function  of  the  second  degree  ^  ( x-^,  x^, 
. . .  .x^  may  be  expressed  as  an  aggregate  of  squares  of  linear 
functions  of  the  variables. 

11.  In  the  proof  of  the  above  theorem  we  assume  that 
<^  (yXy,  X2, .  . .  .x„)  cannot  be  expressed  as  a  function  of  n — 1  vari- 
ables; it  foUows,  therefore,  that  the  inequality 

[20]  2±^"^22----A,„^o 

is  true,  and  it  is  not  possible  to  determine  constants  k,  so  that  the 

equation  ^  k,-  ^^=0  exists  identically. 


*See  also  Lagrange,  Misc.  Taur.,  I.,  p.  18,  17S9 ;  Mecanique,  T.  I.,  3;  Gauss,  Disq. 
Arithm.,  p.  271;   Jheoria  Comb.  Observ.  p.  31,  etc. 


of  Functions  of  Several  Variables.  45 

If,  then,  y  represents  a  linear  function  of  x  having  the  form 

[21]  y  =  C^X^\CT.X^^ +<^n^n, 

and  if  ^  is  a  certain  constant,  then  the  expression  <^ — g  y^,  after 
the  theorem  proved  above,  can  be  expressed' as  a  function  of  only 
n — 1  variables,  if  the  constants  k^,  k2,....k„  may  be  so  deter- 
mined that 


\=i 


or, 


[22]  ^K4>x-g}'^hc^=o- 

\=l  X=l 

From  the  assumption  made  regarding  [20] ,  it  follows  on  the 
one  hand  that  the  inequality 


[23] 


2^: 


Cx^O 


\  ^\< 


must  exist.     On  the  other  hand,  in  virtue  of  the  n  linear  equa- 
tions, 


[24] 


2^'^A'^/*=*^x' 


(X=l,  2,...;^) 


the  quantities  x^,  X2,....x„  may  be  expressed  as  linear  functions 
of  (f>i,  (f)2,  ....(()„ ,  and  consequently  by  the  substitution  of  these 
values  oi  Xi,  X2, . . . . Xa  in  [21]  y  takes  the  form 

[25]  >'=^e^<f>., 

where  the  e,  are  constants,  which  are  composed  of  the  constants 
A^X  and  c\ . 

iBut  fiTom  equation  [22]  it  follows  that 


y 


2  ^^  'f^^ 
^2  ^^  ^^ 


46  Theory  of  Maxima  and  Minima 

Such  a  representation  of  the  <^x'  however,  since  we  have  to  do 
with  linear  equations,  can  be  effected  only  in  one  way. 


We  therefore  have  J^'=^^=J ^"^^^(^^^ 


from  which  it  follows  that 


lt=n 


K=g^x^^^c^     (X=:l,  2,....«). 
Through  the  substitution  of  these  values  in  [22]  we  have 

X=l  X=l 

consequently,  owing  to  the  relation  [25] ,  we  have 

x=« 

X=l 

This  value  of  g-  may  be  expressed  in  a  different  form,  for  from 
[25]  and  [17]  it  follows  that 

»=«  »=« 

I/=l  y=l 

Comparing  this  result  with  [21],  we  have 

[27]  c,  =(^,(e„  ej, e„)  (v=l,  2,....n), 

and  consequently, 


[28] 


or,  from  [18] 


of  Functions  of  Several  Variables.  47 

1 


x=w 


2^'^*^^  (^1'  ^^2' — ^") 


\=\ 


(j)  (ei,  e^, e„) 


Since  the  quantities  Cj,  Cj, . . . .  c„  are  perfectly  arbitrary  ex- 
cept the  one  restriction  expressed  by  the  inequality  [23],  the 
quantities  ei,  e2,....e„  are  in  consequence  of  the  equation  [27] 
completely  arbitrary  with  the  one  limitation  resulting  from  [28], 
viz.,  the  function  <f)  cannot  vanish  for  the  system  of  values 
(ei,  e^,. . .  .€„):  otherwise  ^  would  become  infinite. 

12.  Reciprocally,  if  the  quantities  e^,  gj, . .  . .  e„  are  arbitrarily 
chosen,  but  with  the  restriction  just  mentioned,  and  if  g  is  deter- 
mined through  [28],  it  may  be  proved  that  the  expression 
^ — •,^>'^  where  ;>'  has  the  form  [25],  may  be  expressed  as  a  func- 
tion of  only  n — 1  variables. 

For  form  the  derivatives  of  this  expression  with  respect  to 
the  different  variables,  and  multiply  each  of  the  resulting  quanti- 
ties by  the  constants  e^,  e-i,....  e„.  Adding  these  products  and 
having  regard  to  [28],  we  have 

^e\'^\—gy  ^ex<^\  (^1.  ^2 en)  =  ^e\^\—y- 

The  expression  on  the  right-hand  side  is  zero  from  [25].  Hence 
we  may  choose  n  constants  in  such  a  way  that  the  sum  of  the 
products  of  these  constants  and  the  derivatives  of  the  expression 
^ — gy^  is  identically  zero.  From  this  it  follows  by  a  similar 
method  of  reasoning  as  was  given  in  connection  with  the  equation 
[14]  that  the  expression  ^ — gy^  may  be  expressed  as  a  function 
of  only  n — 1  variables. 

13.  If  we  represent  ^  (x^  x^, . .  .x,,)—gy^  by  ^  {x^,  x^, x„), 

we  may  derive  this  function  of  n — 1  variables  from  [19]  by  sub- 
stituting x\-yte\  for  x\  (X==l,  2, . . .  .«)  in  ^;  if  one  of  these 
arguments  is  made  equal  to  zero,  we  have  as  in  art.  9 


48  Theory  of  Maxima  and  Mimvia 

<f>  (Xi,  Xi Xn)—gy=<l>  (^1— — ^k, ^k ^^^^=^^k,  O,  Xi,+i  — 

^k+1  ^  „  ^n^     \ 

-t-k)  •  •  •  •  -t-n  't'k  l> 

^k  ^k        / 

f 

or,  if  the  new  arguments  are  represented  by  x-^,  x'2, ....  x\-i, 

^  (Xi,  X2, x„)—g  x^^  (f>  (x\,  x'2, x'„^i). 

Employing  the  same  method  of  procedure  with  ^  {x\,  x'2 ^'„-i) 

as  was  done  with  (f>  {xi,  X2,...  .x„),  we  come  finally  to  the  function 
of  only  one  variable,  which  being  a  homogeneous  function  of  the 
second  degree  is  itself  a  square.  Hence  we  have  the  given  homo- 
geneous function  <^  (x^,  X2, . . .  .x„)  expressed  as  the  sum  of  squares 
of  linear  homogeneous  functions  of  the  variables.  If  the  coeffic- 
ients of  <f)  are  real  as  also  the  quantities  e,  the  coefficients  g  are 
also  real  and  since  the  quantities  e  may  with  a  single  limitation 
be  arbitrarily  chosen,  it  follows  that  a  transformation  of  such  a 
kind  that  the  result  shall  be  a  real  one,  may  be  performed  in  an 
infinite  number  of  ways.* 

14.    If  now  the  expression 

[29]  <j>  (x„  X2,....  x^=g^y^  +  ^2  >'2H  . .  • .  +  gny^ 

is  to  be  continuously  positive  for  real  values  of  the  variables  and 
equal  to  zero  only  when  the  variables  themselves  all  vanish,  then 
all  the  qualities  g^  gv-  ■  •  -gn  must  be  positive  ;  for  if  this  was  not 
the  case,  but^i  (say)  was  negative,  then  since  the  jv's  are  inde- 
pendently of  each  other  linear  homogeneous  functions  of  the  ;*;'s 
we  could  so  choose  the  x's  that  all  the  jy's  except  jFi  vanished  and 
consequently  contrary  to  our  assumption  (f>  {xi,  X2,....x„)  would 
be  negative.  Further  none  of  the  ^'s  can  vanish  ;  for  if  £^i,  say, 
was  zero,  we  might  so  choose  a  system  of  values  x^,  X2,....Xn  in 
which  at  least  not  all  the  quantities  x^,  X2,....x^  were  zero  that 
all  the  ^-'s  vanished  except  y-^,  and  consequently  </>  could  then  be 
zero  without  the  vanishing  of  all  the  variables  x^,  Xj,....  x„. 


^  See  Burnside  and  Panton,  Theory  of  Equations.     1892.     Page  430. 


of  Functions  of  Several  Variables. 


49 


Reciprocally  the  condition  of  ^i,  gz,...  ■£„  being  all  positive  is 
also  sufficient  that  <f>  be  continuously  positive  for  real  values  of 
the  variables,  and  that  <^  be  equal  to  zero  only  when  all  the  vari- 
ables vanish. 

15.  In  order  to  have  in  as  definite  a  form  as  possible  the  ex- 
pression of  <^  as  a  sum  of  squares,  we  shall  give  to  the  expression 
[26]  for  g  still  a  third  form. 

In  connection  with  [12]  it  follows  from  [27]  that 


/*=« 


'=2^.M^^  ("=!'  2 n). 


c„  = 


M=l 


Denote  by  A  the  determinant  of  these  equations,  which  from  [20] 
is  not  identically  zero,  i.  e., 

[30]  A  =  ^±  AnA^....A„„. 

We  have  as  the  solution  of  the  preceding  equation 


''  =  A 


^2 


dA 


dA 


x=i 


\^i 


cx  (/^=1,  2, n). 


It  follows  from  this  in  connection  with  [26]  that 

A 


[31] 


g 


dA 


\,ii 


dA 


c\    Cu 


xm 


an  expression  in  which  the  c's  are  subject  only  to  the  one  condition 

dA 


that 


^1  ^~2 — ^^  ^/^     IS  not  identically  zero. 


16.  It  shall  next  be  shown  that  we  may  separate  from 
(f)  {xi,  X2,....  x„)  the  square  of  a  single  variable  in  such  a  way  that 
the  resulting  function  contains  only  n — 1  variables. 

For  example,  in  order  that  the  expression  (ji — g  x„^  be  ex- 
pressed as  a  function  of  n — 1  variables,  we  have  chosen  for  g  the 
value  [31],  after  we  have  written  in  this  expression  c\—o  (X  —  1, 
2, . . . .  n — 1),  while  to  c„  is  given  the  value  unity. 


so  Theory  of  Maxima  and  Minima 

From  this  we  have 


dA  A^ 


9A„ 

where  A-^  is  the  determinant  of  the  quadratic  form  ^{^x^,  X2,.... 
x„_i,o).     Of  course,  this  determinant  must  be  different  from  zero. 
Hence  we  may  write 

where 

'^(x\,  x'2, ....  x'„^i)  =4>  (^1——  ^„,  X2—^X^,  ....  X„_i—^^=^X„,  oj. 

We  may  then  proceed  with  (f>  just  as  has  been  done  with  (f)  by 
separating  the  square  of  x'^^i,  etc. 

If  we  notice  that  the  determinant  of  the  the  function  of  n — /a 
variables  which  results  from  the  seperation  of  /i  squares  is  the 
same  as  the  determinant  of  the  function  which  results  from  the 
original  function,  when  we  cause  the  fi  last  variables  to  vanish  (in 
this  function),  and  if  we  denote  this  determinant  by  ^^,  we  have 
the  following  expression  for  <^  : 

17.   If  now  <!>  is  to  be  continuously  positive  and  equal  to  zero 
only  when  all  the  variables  vanish,  the  coefficients  on  the  right- 
hand  side  of  the  above  expression  must  all  be  greater  than  zero. 
We  therefore  have  the  theorem : 
In  order  that  the  quadratic  form 

<p  {Xi,  X2,  .  .  .  .X„)  =  ^^^X/i^fi^M    ^X;u  =  ^^x> 

A, /t 

de  a  definite  form  and  rem,ain  continuously  positive,  it  is  neces- 
sary and  sufficient  that  the  quantities  A^,  A2,. . .  .^,,-1,  which 

are  defined  through  the  equation  A^  =  ^  —  ^u  ^22- . .  -^n-^,  n-« 


of  Fundiotis  of  Several  Variables.  51 

be  all  positive  and  different  from  zero.     If  on  the  other  hand 
the  qardratic  form  is  to  rem,ain  continuously  negative,  then  of 

the  quantities  A^-i,  ^„-2, A^,  A,  the  first  must  be  negative, 

and  the  following  must  be  alternately  positive  and  negative. 


APPLICATION  OF  THE  THEORY  OP  QUADRATIC  FORMS  TO  THE 
PROBLEM  OF  MAXIMA  AND  MINIMA  STATED  IN  ARTS.  1 — 6. 

18.  By  establishing  the  criterion  of  the  previous  article  the 
original  investigation  regarding  the  maximum  and  minimum  of  the 

function /(;t;i,  x^, x^  is  finished.     The  result  established  in  art. 

12  may  in  accordance  with  the  definitions  given  in  art.  7  be  ex- 
pressed as  follows :  in  order  that  a  maxim^um  and  minimum,  of 
the  function  f  {^x-^,  x^,...  .x,^  may  in  reality  enter  on  the  position 
(«!,  fl5i, . . .  .a„)  which  is  determined  through  the  equations  [6], 
it  i^  necessary,  if  the  second  derivatives  of  the  function  do  not 
all  vanish  at  this  position,  that  the  aggregate  of  the  term,s  of 
the  second  degree  of  the  equation  [4]  be  a  DEFINITE  quadratic 
form.  We  can  more  especially  say,  as  already  indicated  in  art.  8: 
If  we  have  a  definite  form  which  only  vanishes  when  all  the 
variables  vanish,  the  function  has  on  the  position  in  question 
in  reality  a  maximum  or  m^inimum  value;  if,  however,  the 
fortn  vanishes  for  other  values  of  the  variables,  then  a  deter- 
mination as  to  whether  a  maximum,  or  minimum-  in  reality  ex- 
ists, is  not  effected  in  the  fnanner  indicated  and  requires 
further  investigation  as  is  seen  below. 

In  accordance  with  the  theorem  stated  in  art.  8,  a  maximum 
or  minimum  will  enter  for  a  system  of  real  values  of  the  equation 
[6],  if  the  homogeneous  function  of  the  second  degree 

-S://9'/(^i,  ^2.----^n)'j    h^h^     \ 


\/* 


X,  M 


is  a  definite  quadratic  form  ;  that  is  (art.  17)  there  will  be  a  mini- 
mum on  the  position  {tZi,  a^,. . . .  a„)  if  the  quotients 


hear 

y  of  Maxima  and  Minima 

F, 

F, 

F.-^ 

F,' 

F^  

F.-,' 

52 


where  7^^  =  ^  ±/ii  y^2+ /^n-^,  n-/x ,  ^^^  all  positive,  a  maxi- 
mum, if  they  are  all  negative;  and  if  in  both  cases  the  quotients 
are  different  from  zero. 

This  last  condition  is  only  another  form  of  what  was  said 

above,  viz.,  that    ^/^m  ^^  ^/^    cannot   be   a   definite   quadratic 

form,  which  vanishes  for  other  values  of  the  variables  that  are  not 
all  zero.     This  condition  is  in  general  necessary.      For,  if  (say) 

then,  the  summation  ^^fk,i.  ^K  ^^  being  denoted  by  ^{h^,  h^,,.. 

. .  ^„),  this  equation  would  directly  imply  the  existence  of  a  rela- 
tion of  the  form  : 


2  ^Av\K  h^., h^)^o, 


where  the  ^i,are  constants  which  do'not  all  simultaneously  vanish. 
If,  therefore,  k^  (say)  is  different  from  zero,  we  may  write 

"=1 
and  with  the  help  of  this  relation  we  have  from  the  equation 

X=«  \=n— 1 

<^(Ai,  h,,....K)^^^^(^k„h,,....h„)  h^=^,K+  ^<f>xhx 
the  following  relation 

X=u— 1  y 

Now  in  this  expression  we  may  so  choose  the  arbitrary  quantities 
h,  that 


of  Functions  of  Several  Variables. 


S3 


h   ——h 


(\=l,2,....w), 


and  consequently  the  function  ^  (/^j,  h^,....  A„)  would  vanish  with- 
out all  the  A's  becoming  simultaneously  zero. 

Hence  with  this  system  of  values  of  the  A's  the  difference 
f{xx,X2, . .  .x„) — /{au  az,--.  .«„)  would  begin  with  terms  of  the 
third  dimensions;  and  consequently  the  function /"(;»;i,  X2,....x„) 
would  have  no  maximum  or  minimum  on  the  position  {a^,  Uj,...  .a„) 
unless  all  the  terms  of  the  third  dimension  vanish,  a  further  con- 
dition being  also  that  the  aggregate  of  the  terms  of  the  fourth 
dimension  have  a  continuously  negative  or  a  continuously  posi- 
tive sign.  y 

We  are  not  able  to  give  the  criterion  for  this;  it  is,  however, 
the  more  improbable  that  such  a  case  happens,  the  greater  the 
number  of  variables  that  appear,  since  the  necessary  conditions 
(that  the  terms  of  the  third  dimension  shall  all  vanish)  continu- 
ously increase. 

The  problem  of  this  chapter  is  thus  completely  treated;  how- 
ever the  conditions  that  a  quadratic  form  shall  be  a  definite  one, 
appear  in  a  less  symmetric  form  than  we  wish.  It  is  due  to  the 
fact  that  we  have  given  special  preponderance  to  certain  variables 
over  the  others. 

We  shall  consequently  take  up  the  same  subject  again  in  the 
next  chapter. 


54  Theory  of  Maxima  and  Minima 


CHAPTER  III. 

THEORY   OP   MAXIMA   AND   MINIMA   OF   FUNCTIONS    OF    SEVERAL 
VARIABLES   THAT   ARE   SUBJECTED   TO    SUBSIDIARY  CONDITIONS. 

1.  In  the  preceding  investigations  the  variables  x-^,  x^,.  . .  .x^ 
were  completely  independent  of  each  other. 

We  now  propose  the  problem  :  Among  all  systems  of  val- 
ues (Xi^,  X2,...  .x^  find  those  which  cause  the  function  F(xi,  X2, 
. .  .  .Xn)  to  have  maximtt7n  and  minim^um  values  and  which 
at  the  same  tim-e  satisfy  the  equations  of  conditions  : 

[1]  /x(^i.  ^2. ^n)  =  o    (X=l,  2, m;  m<n), 

where  f  {x^,  X2,  ■  ■  ■ .  ^„)  are  functions  of  the  sam-e  character  as 
fix^,,  X2,...  .x„)  in  Chapt.  11. ,  art.  2. 

2.  The  most  natural  way  to  solve  the  problem  is  to  express 
by  means  of  equations  [1]  m  of  the  variables  in  terms  of  the  re- 
maining n — m  variables  and  write  their  values  in  F{xi,  X2,...  -^n). 
This  function  would  then  depend  only  upon  the  n — m  variables 
which  are  completely  independent  of  one  another,  and  so  the  pre- 
sent problem  would  be  reduced  to  the  one  of  the  preceding 
chapter. 

This  method  of  procedure  in  practice,  however,  cannot  be 
performed  in  most  cases,  since  it  is  not  always  possible  by  means 
of  equations  [1]  to  represent  in  reality  ^w  variables  as  functions  of 
the  n — m-  remaining  variables.  We  .must  therefore  seek  a  more 
practicable  method. 

3.  If  («!,  a2, .  . .  .  a^)  is  any  system  of  values  of  the  quantities 
Xi,  X2, .  . .  .Xn  which  satisfy  the  equations  [1],  then  of  the  systems 
of  values  in  the  neighborhood  of  the  position 

(x^=ai  +  hi,  X2^a2+h2, ;r„=a„  +  ^„), 

we  can  consider  only  those  which  satisfy  the  equations    [1],  for 
which  therefore  we  have 


of  Functions  of  Several  Variables.  55 

[2]        f\{,a^^ K  a^^  K at„  +  ^„)=-o,  (X=l,  2, m). 

Hence  by  Taylor's  theorem  the  A's  satisfy  the  equations 

(X=l,  2,....w) 

where   \h^,  h^,,....  A„]^  denotes  the  terms  of  the  second  and  higher 

dimensions. 

4.  It  being  assumed  that  at  least  one  of  the  determinants  of 
the  wth  order,  which  can  be  produced  by  neglecting  m — n  col- 
umns from  the  system  oi  m-  n  quantities 

y  11    1  y  12    '      /in  ) 

r^l  /       7  21    '  y  22    t      y  2n  t 

/mliym2>      ymn> 

is  different  from  zero,  then  (see  Chapt.  I.,  arts.  14  and  15)  m  of  the 
quantities  h  may  be  expressed  through  the  remaining  n — m  quan- 
tities (which  may  be  denoted  by  k^,  ^2- •  •  •  •'^n-m)  in  the  form  of 
power-series  as  /bllows  : 

|_5J         A^x  =  \K\i  A^2>  •  •  ■  •  l^a-m)  y^   +  V^l)  ^2)  •  •  •  •  "^n-m  ^  ^  ~^ ' 

(\=1,  2,...w) 

where  the  upper  indices  denote  the  dimensions  of  the  terms  with 
which  they  are  associated.  These  series  converge  in  the  manner 
given  in  Chapt.  I.,  art.  15;  they  satisfy  identically  the  equations 
[2]  and  furnish,  if  the  quantities  k^,  k-j,,....  ^„_„  are  taken  suf- 
ficiently small,  all  values  of  the  m  quantities  h  which  satisfy  these 
equations. 

5,  In  accordance  with  Chapt.  I.,  art.  14  one  of  the  determi- 
nants of  the  ^wth  order  of  the  system  [4]  must  be  different  from 
zero,  in  order  that  the  considerations  of  the  preceding  article  be 
true.  This  condition  is  in  general  satisfied ;  there  are,  however, 
special  cases  where  this  is  not  the  case.  A  geometrical  interpre- 
tation will  explain  these  exceptions. 

Let  F  and  an  equation  of  condition, /"=  <?  contain  only  three 
variables  x.^^ ,  x-i  and  x^ 


56  Theory  of  Maxima  and  Minima 

The  equation  of  condition  f^x^-,  ^21  x%)^o  represents  then  a 
surface  upon  which  the  point  {^x-^,  x^,  X;^,  having  the  orthagonal  co- 
ordinates Xi,  X2  and  X2,  is  to  lie,  and  for  which  F{xi,  X2,  X3)  is  to 
have  a  maximum  or  minimum  value. 

The  determinants  of  the  first  order  in  the  development 

/(ai  +  Ai,  a^+hi,  ^3  +  ^3)— /(«i.  «2.  ^3) 
with  respect  to  powers  of  h^,  h-^  and  h^  cannot  all  be  equal  to  zero; 
that  is,  all  the  terms  of  the  first  dimension  cannot  vanish,  the 
single  terms  being  these  determinants ;  and  this  means  that  the 
surface /"=  (3  cannot  have  a  singularity  at  the  point  in  question. 

Take  next  two  equations  of  condition  fx'=o  and  fi.=o  be- 
tween three  variables  x-^.,  x-i  and  x^,.  Considered  together,  they 
represent  a  curve  and  the  condition  that  the  corresponding  deter- 
minants of  the  second  order  cannot  all  be  zero,  means  here  that 
the  curve  at  the  point  in  question  cannot  have  a  singularity. 

6.  We  may  therefore  assume  that  a  determinant  of  the  m'Ca. 
order  of  the  system  [4]  is  different  from  zero.  Thus  the  develop- 
ments [5]  are  possible.  These  power-series  satisfy  identically 
the  equations  [2]  and  offer,  if  the  quantities  k^,  k^i- . . . ^„-m  are 
taken  sufficiently  small,  all  the  values  of  the  m  quantities  h\  which 
likewise  do  not  exceed  certain  limits.  Suppose  that  the  values  of 
the  m  quantities  h\  are  substituted  in  the  difference 

F{x^,  X2, Xn)  —  F{au  ^2, «„), 

an  expression  which  then  depends  only  upon  the  n — m  variables 
^1,  ^2'  •  •  •  •  ^n-m  that  are  independent  of  one  another,  and  may  con- 
sequently for  sufficiently  small  values  of  these  variables  be  de- 
veloped in  the  form 

[6]  F{xy,X2, x„)  —  F(^ai,a2, a„) 

p=i  p,<' 

We  saw  (Chapt.  II.,  art.  6)  that  in  order  to  have  a  maximum  or 
minimum  on  the  position  (^j,  a^,...  .fl^n).  it  is  necessary  that  the 
terms  of  the  first  dimension  vanish,  and  consequently 

[7]  e^=o     (p=l,2,....n—m). 


of  Functions  of  Several  Variables. 


57 


7,     This  condition  may  be  easily  expressed  in  another  manner: 
For  we  obtain  the  quantities  e,  if  in  the  development 


2^M  K-^-^^^y^'   ^y.  ^^  + 


^=1  /i,  V 

we  substitute  in  the  terms  of  the  first  dimension  the  values  of  the 
the  m  quantities  from  [5]  and  arrange  the  result  according  to  the 
quantities  k^,  k^,. . . . ^„-m.     In  other  words,  the  equations  [7]   ex- 

press  the  condition  that  ^  F^^  h^  must  vanish  identically  for  all 

systems  of  values  of  the  ^'s,  that  satisfy  the  tn  equations  [3] 
after  they  have  been  reduced  to  their  linear  terms.  These  are  the 
m  equations 


M=« 


[8] 


2A.  K  =  o   {X=l,2,....m). 


/t=l 


Now  multiply  the  m  equations  respectively  by  the  m  arbitrary 
quantities  e^,  gjt  •  •  •  -^m.  and  adding  the  results  to  the  equation 

we  have  the  following  equation : 


^=1 


But  we  may  so  determine  the  e's  that  those  terms  in  this  sum- 
mation drop  out  which  contain  the  in  quantities  h,  which  are  ex- 
pressed in  [5]  through  the  n — m  other  h'^ ;  by  causing  these 
terms  to  vanish  we  have  a  system  of  m  linear  equations,  whose 
determinant  by  hypothesis  is  different  from  zero. 

Since  the  terms  which  remain  of  equation  [9]  are  multiplied 
by  the  completely  arbitrary  quantities  ki,  ^2>  •  •  •  •  ^n-ra>  it  is  not 
possible  for  this  equation  to  exist  unless  each  of  the  single  co- 
efficients is  equal  to  zero. 


58  Theory  of  Maxima  and  Minima 

Consequently  we  have  as  the  first  necessary  condition  for  the 
appearence  of  of  a  maximum  or  minimum  the  existence  of  the 
following  system  of  n  equation  : 

and  indeed  in  the  sense  that  if  m  of  these  equation  exist  independ- 
ently of  one  another,  the  remaining  n — m  of  them  must  be  identic- 
ally satisfied  through  the  substitution  of  the  e's  which  are  de- 
rived from  the  first  m  equation,  it  being  of  course  presupposed 
that  the  system  of  values  (^i,  ^2, ....a„)  has  already  been  so 
chosjen  that  the  equations  [1]  are  satisfied. 

Taking  everything  into  consideration  we  may  say  :  In  order 
that  the  function  I^  (xi,  X2,  ■  ■  ■ .  x^)  have  a  maximum  or  m>ini- 
mum-  on  any  position  (a^,  a^.,. . .  .a„),  it  is  necessary  that  the 
n-\-m  equations 


1^..  aA..lA  + +..,|A=», 


[10] 


dx^          dx^          dxfi        ' '             "'  dx^ 
/\(xi,  X2, x„)  =  0      (X=l,  2, m), 


be  satisfied  by  a  system  of  real  values  of  the  n-^m.  quantities 

8.  These  deductions  are  connected  with  the  one  assumption 
that  at  least  one  of  the  determinants  of  the  m^th.  order  which  can 
be  formed  out  of  the  m- .  n  quantities  [4]  through  the  omission  of 
n — m.  columns  does  not  vanish.  This  condition  was  necessary 
both  for  the  determination  of  the  quantities  h,  which  satisfy  the 
equations  [2]  and  also  for  the  determination  of  the  m^  factors  e^. 

It  may  happen  that  a  maximum  or  minimum  of  the  function 
/^enters  on  the  position  {a^,  aj, . . .  .a„)  even  when  the  above  con- 
dition is  not  satisfied.  For  if  it  is  possible  in  any  way  to  deter- 
mine all  systems  of  values  of  the  A's  not  exceeding  certain  limits 
that  satisfy  the  equations  [2],  the  equations  [7]  together  with  the 
equations  [1]  are  sufficient  in  number  to  determine  the  n  quanti- 
ties «!,  a-i,.  ■  ■  .a^. 

When  the  above  condition  is  not  satisfied,  the  equations  [8] 
exist  identically  and  consequently  the  equations  [3] ,  which  serve 


of  Functions  of  Several  Variables.  S9 

to  determine  the  A's  begin  with  terms  of  the  second  dimension. 
We  may  often  in  this  case  proceed  advantageously  by  introducing 
in  the  place  of  the  original  variables  a  system  of  n — m  new  vari- 
ables so  chosen  that  when  they  are  substituted  in  the  given 
equations  of  condition,  they  identically  satisfy  them. 

9.  To  make  clear  what  has  been  said,  the  following  example 
will  be  of  service ;  its  general  solution  is  given  in  the  sequel 
(Chapt.  III.,  art.  28). 

The  problem  proposed  is :  find  the  shortest  line  which  can  be 
drawn  from  a  given  point  to  a  given  surface.  Upon  the  surface 
there  are  certain  points  of  such  a  nature  that  the  lines  joining 
these  points  with  the  given  point  have  the  desired  property  and 
besides  stand  normal  to  the  surface  at  these  points. 

If  by  chance  it  happens  that  one  of  these  points  is  a  double 
point  (node)  of  the  surface,  so  that  at  it  we  have  f  =  o,  f=^o, 
fz=o,  then  in  reality  for  this  point  the  terms  of  the  first  dimension 
in  the  equations  [2]  drop  out  and  we  have  the  case  just  mentioned. 

If  the  surface  is  the  right  cone : 

fi^x,  y,  z^~  o=x'^+y — z^, 
then  we  may  write  : 

I     x=2uv, 
[11]  <    y=u'—v\ 


\ 


---u^  +  zJ^. 


The  equation  of  the  surface  is  identically  satisfied,  and  it  is  easily 
seen  that  we  may  express  the  quantities  hi,  h-^,  h^,  through  two 
quantities  k^  and  ^2  independent  of  each  other  even  in  the  case 
where  the  required  point  of  the  surface  is  the  vertex  of  the  cone, 
that  is,  the  point  x=o=y=z,  or  u—o=v.  This  representation 
may  indeed  be  effected  in  such  a  way  that  not  only  infinitely  small 
values  of  h^,  hj,  h^  correspond  to  infinitely  small  values  of  k^,  k^, 
but  also  that  all  systems  of  values  h^,  hj,  h^  are  had  which  satisfy 
the  equation 

f{x+hi,y  +  h^,  2+h:i)=o. 

We  have,  however,  to  give  to  the  variables  at  one  time  real,  at 
another  time  purely  imaginary  values,  if  the  equation  [11]  is  to 
represent  the  entire  surface  of  the  cone ;    but  in  this  manner  the 


60  Theory  of  Maxima  and  Mitmna 

unavoidable  trouble  has  taken  such  a  direction  that  the  proposed 
problem  falls  into  two  similar  parts,  which  may  be  treated  in  full 
after  the  methods  of  Chapt.  II.  In  other  cases  we  may  pro- 
ceed in  a  similar  manner.  The  special  problem  will  each  time  of 
itself  offer  the  most  propitious  method  of  procedure. 

10.  We  must  now  establish  the  criteria  from  which  one  can 
determine  whether  a  maximum  or  minimum  of  F{xi,  X2,....x„) 
really  enters  or  not  on  a  definite  position  (ui,  ^2, .  .  .  .«„),  which  has 
been  determined  in  accordance  with  the  theorem  cited  in  art.  7  of 
the  present  chapter. 

One  might  consider  this  superfluous,  since  in  virtue  of  the  cri- 
teria given  in  the  previous  chapter  a  maximum  or  minimum  will 
certainly  enter,  if  the  aggregate  of  terms  of  the  second  dimension 
in  [6]  is  a  definite  quadratic  form  of  the  nature  indicated. 

It  is,  however,  desirable  to  determine  the  existence  of  a  maxi- 
mum or  minimum  without  having  previously  made  the  develop- 
ment of  the  function  in  the  form  [6] ;  since,  in  order  to  obtain  the 
coefficients  Cp  <r ,  we  must  pay  attention  not  only  to  the  terms  of  the 
first  dimension  but  also  to  the  terms  of  the  second  dimension, 
when  the  values  of  [5]  are  substituted  in  the  development  of 

F(xi,  X2, x„)—F(ai,  a., a„)  =  ^F^^  k^,+ 

-—2^"''  h),hv^ 


M=l 


At,  V 


11.  The  above  difficulty  may  be  avoided,  if  we  multiply  each 
of  the  expressions  [2]  which  vanish  identically  by  the  quantities 
e„(jLt=l,  2, . . .  .in)  respectively,  add  them  thus  multiplied  to  the 
above  difference  and  then  develop  the  whole  expression  with  re- 
spect to  the  powers  of  h. 

Owing  to  equation  [9]  terms  of  the  first  dimension  can  no 
longer  appear  in  this  development,  and  we  have,  if  we  write 


M=m 


[12]  /^+2^m/m=^, 


/i=l 


of  Functions  of  Several  Variables. 
[13]       F{x^,  X2, x„)—F{au  Uj, a^  =  G  {x^,  x^, x,^ 


61 


-G  (ui,  Ui, a„)  =  -j^  G^ 


fX  V    '^y.     '*^  V 


+ 


M, " 


We  thus  have  the  homogeneous  function  of  the  second  degree 
2  e^„  k^  kg  of  the  formula  [6],  if  we  substitute  in  ^\G^^h^  h„ 


/>. " 


M,  » 


the  values  [5]  and  consider  only  the  terms  of  the  first  dimension 
in  the  process.  If  we  then  apply  the  criteria  of  the  preceding 
chapter,  we  can  determine  whether  the  function  F  possesses  or 
not  a  maximum  or  minimum  on  the  position  (a„  a^, . .  -cia)- 

12.  The  definite  conditions  that  have  been  thus  derived  are 
unsymmetric  for  a  two-fold  reason :  on  the  one  hand  because  in 
the  determination  of  the  quantities  h  some  of  them  have  been 
given  preference  over  the  others,  and  on  the  other  hand  because 
those  expressions,  by  means  of  which  it  is  to  be  decided  whether 
the  function  of  the  second  degree  is  continuously  positive  or  con- 
tinuously negative,  have  been  formed  in  an  unsymmetric  manner 
from  the  coefficients  of  the  function. 

It  is  therefore  interesting  to  derive  a  criterion  which  is  free 
from  these  faults,  and  which  also  indicates  in  many  cases  how  the 
results  will  turn  out.  With  this  in  view  let  us  return  to  the  prob- 
lem already  treated  in  the  preceding  chapter  and  propose  the  fol- 
lowing more  general  theorem  in  quadratic  forms. 


THEORY  OP  HOMOGENEOUS  QUADRATIC  FORMS. 

13.      Theorem.      We  have  given  a  homogeneous  function  of 
the  second  degree 


[14] 


<p  yx^,  X2,  • .  .Xn)  —  ^^  A^^Xx  x^ 


\,fi 


y^Xjx  —  ^/n  \  ) 


in  n  variables,  which  are  subjected  to  the  linear  homogeneous 
equations  of  condition 

[15]  d-^-^^^a^^  x^=^o    (X=l,  2, w;  m<n)\ 


62  Theory  of  Maxima  and  Minima 

we  are  required  to  find  the  conditions  under  which  <j)  is  contin- 
uously positive  or  continuously  negative  for  all  those  systems 
of  values  of  the  variables  which  satisfy  equations  [15]. 

It  is  in  every  respect  sufficient  to  solve  this  theorem  v^ith  the 
limitation  that  the  quantities  ;*:  are  subjected  to  the  further  con- 
dition : 

[16]  x^-\-xi^  ....^rxl=\\ 

since,  if  <^  is  a  definite  quadratic  form  and  positive^  sa)%  it  is  also 
a  definite  positive  form  for  those  values  of  x-^,  X2,  ■  ■  ■  .x„  v^^hich 
satisfy  the  equation  [16]. 

Reciprocally,  if  <f)  (x^,  x^, . . .  .x^)  is  positive  for  all'  the  argu- 
ments vv^hich  satisfy  the  equation  [16],  then  it  is  also  positive  for 
all  arbitrary  arguments. 

For  let  Xi,  x^^-.-.x^  be  an  arbitrary  system  of  values  and  form 
the  expressions 

6=—^=%=====     (.=1,2,....^). 

V  x^^  +  x^+  ....  +xl 

Noting  that  f,(«=l,  2,....n)  satisfy  the  equation  [16]  and  writ- 
ing 5=  V x^-^x^-\- +  ;i:^,  it  is  seen  that 

Hence  <f>(xi,  x^,. . .  .x„)  is  positive  when  ^  (fi,  ^2'  •  •  •  •  fn)  is  positive. 

It  is  therefore  in  every  respect  admissible  to  add  the  equation 
[16].  We  have,  however,  thereby  gained  an  essential  advantage: 
for  owing  to  the  condition  [16]  none  of  the  variables  can  lie 
without  the  interval  — 1....  +1;  further  since  the  function 
varies  in  a  continuous  manner,  it  must  necessarily  have  an  upper 
and  a  lower  limit  for  these  values  of  the  variables  x^,  X2,....x„ ; 
that  is,  among  all  systems  of  values  which  satisfy  the  equations 
[15]  and  [16]  their  must  necessarily  be  one  for  which  there  is  a 
maximum  and  one  which  gives  a  minimum  value  of  (j). 

We  limit  ourselves  to  the  determination  of  the  latter. 

14.  Through  the  addition  of  equation  [16]  we  have  reduced 
the  theorem  of  the  preceding  article  to  a  problem  in  the  theory  of 
maxima  and  minima ;  for,  if  the  minimum  value  of  <^  (xi,  x^, . . 
.  .Xa)  is  positive,  <}>  is  certainly  a  definite  positive  form. 


of  Functions  of  Several  Variables. 


63 


If  we  write 


[17] 


\=1  p=l 


then,  in  order  to  find  the  position  at  which  there  is  a  minimum 
value  of  the  function,  we  have  to  form  the  system  of  equations 


dG 


This  gives 


o      (X=l,  2,. . .  -n). 


P=m 


3^ 


-2exx+2ya^ 
p=i 


^=0, 


(\=1,  2,..../?) 


or, 

[18] 


11=11 


P=m 


2  '-^X/.  ^/^— ^  ;i;a.  +  2  ^P  ^P^  ^  ^• 


/*=i 


p=i 


From  the  w+???+l  equations 


(X=l,  2,....«) 


[19] 


li=n 


P=m 


2^>-/^^M— ^ ^x+  2^'' «px=^.  (^=1' 2,....«) 


*  M=i 


p=i 


ix=n 


2^p^^/^=^'  (p=i> 2,... .w) 


M=l 


X=« 


2-5=1; 


the  «  +  ^w-|-l  quantities  x^,  Xj, . .  .  .x^,  ^i,  ^2.  •  •  •  -^mt  ^  J^ay  be  deter- 
mined. Since  we  know  a  priori  that  a  minimum  value  of  the 
function  <^  in  reality  exists  on  one  position,  we  are  certain  that 
this  system  of  equations  must  determined  at  least  one  real  system 
of  values. 

Consequently  the  first  n^m  linear  homogeneous  equations  of 
[19]  'are  consistent  with  one  another,  and  may  be  solved  with  re- 
spect to  the  unknown  quantities  x^^,  Xj,....  x„,  e„  ^2, . .  . .  e„^ ;  their 
determinant  must  therefore  vanish,  and  we  mvist  have 


64 


Theory  of  Maxima  and  Minima 


[20]  Ae  = 


A.x\       &  ,       -<^12»  •  •  • 
^21  '        ^22        ^' 


-^2ii  t 


^12  > 


•   <3t„2 


a 


ml 


a 


mil 


■  a„ 


o,  . . .  .0 


The  equation  A  e  =  o  is  clearly  of  the  n — mth  degree  in  e.  The 
minimum  value  of  ^  is  necessarily  contained  among  the  roots  of 
this  equation  ;  for  if  we  multiply  the  equations  [18]  respectively 
by  x-^,  X2,....x„  and  add  the  results,  we  have 


[21] 


<f)(xi,X2, x„)=e, 


it  being  presupposed  that  the  system  of  values  (x^,  x^, . .  ■  ■  x„) 
together  with  the  quantities  e^,  e^,....  e^  satisfies  the  system  of 
equation  [19],  which  is  only  possible  if  e  is  a  root  of  the  equa- 
tion A  e=  (3.  Further  among  the  systems  of  values  x  which 
satisfy  the  system  of  equations  [19],  that  system  is  also  to  be 
found  which  calls  for  the  minimum,  and  since  the  value  of  the 
function  which  belongs  to  such  a  system  of  values  is  always  a  root 
of  equation  [20] ,  it  follows  also  that  the  required  minimal  value 
of  ^  must  be  contained  among  the  roots  of  this  equation. 

As  already  remarked  this  minimal  value  must  be  positive  if  <^ 
is  to  be  continuously  positive  for  the  systems  of  values  of  the  x^?> 
under  consideration,  and  from  this  it  follows  that  A  e  must  have 
only  positive  roots.  For  if  one  root  of  this  equation  was  negative, 
then  for  this  root  we  could  determine  a  system  of  values  x^,  x^, . . 
. .  Xn,  ^1,  ^2, . . .  .  ^n,  for  which  as  seen  from  [21]  <^  is  likewise 
negative. 

Hence  in  order  that  (f>  be  continuously  positive  for  all  sys- 
tems of  values  of  the  x's  which  satisfy  the  equations  [15]^  it  is 
necessary  and  sufficient  that  the  equation  A  e  =  o  have  only 
positive  roots* 

The  question  next  arises  when  does  the  equation  A  e  =  <? 
have  only  positive  roots.     It  may  be  answered  in  a  completely 


■See  Zajaczkowski,  Annals  of  the  Scientific  Society  of  Cracow.  Vol.  XII,  1867 


of  Functions  of  Several  Variables. 


65 


rigorous  manner  by  means  of  Sturm's  theorem*;  but  the  investi- 
gation is  somewhat  difficult,  and  the  symmetry,  which  we  es- 
pecially wish  to  preserve,  would  be  lost,  when  we  applied  Sturm's 
theorem. 

For  develop  the  determinant  according  to    powers  of  e   as 
follows  : 


[22] 


"—79,  e"-"-»  +  Bi  e"-™-2 


•+(— l)"~'"^„-m=c7; 


then,  if  all  the  roots  of  this  equation  are  real  and  positive,  the  co- 
efficients ^  must  be  all  positive  and  ^o,  and,  reciprocally,  if  the 
roots  of  this  equation  are  real  and  the  ^'s  are  all  ><?,  then  also 
the  roots  of  the  equation  A  e  =  c?  are  all  positive. 

16.  We  shall  first  show  that  all  the  roots  of  the  equation 
A  e  =0  are  real  for  the  case  where  no  equations  of  conditions  are 
present. 

Equation  [20]  reduces  then  to  the  form  : 


[23] 


-^21  >        ^22        ^  > ^211 


A. 


A 


n2 


.A„„ — e 


o. 


an  equation,  which  is  called  the  "'EQUATION  OF  SECULAR  varia- 
tions "  and  plays  an  important  role  in  many  analytical  investiga- 
tions, for  example,  in  the  determination  of  the  secular  variations 
of  the  orbits  of  the  planets,  as  well  as  in  the  determination  of  the 
principal  axes  of  lines  and  surfaces  of  the  second  degree. f 


•Hermite,  Crelle,  bd.  52,  p.  43 ; 

Serret,  Alg^bre  Sup.  1866,  t.  1,  p.  581 ; 
Kronecker,  Berlin  Monatsbericht,  1873.  Feb. 

t  In  this  connection  the  reader  is  referred  to  : 
Laplace,  M^m.  de  Paris.    1772.    II.,  pp.  293-363  ; 

Euler,  M^m.  de  Berlin.     1749-50  ;     Theoria  motus  corp.  sol.  1765.  Chapt.  V ; 
Lagrang-e,  M^m  de  Berlin.     1773,  p.  108 : 

Poison  et  Hachette,  Journ.  de  I'Ecole  Polytechn.  Cah.  11,  p.  170.  1802; 
Kummer,  Crelle,  bd.  26,  p.  268  ; 
Jacobi,  Crelle,  bd.  30,  p.  46 ;  ^ 

Bauer,  Crelle,  bd.  71,  p.  40  ; 
Borchardt,  Liouv.  Journ.  t.  12,  p,  30 ; 
Sylvester,  Phil.  Mag.  1852.  II.,  p.  138.;    etc. 


66  Theory  of  Mayitma  and  Minima 

17.  Weierstrass'  proof*,  which  is  very  simple,  that  all  the 
roots  of  this  equation  are  real,  depends  only  upon  the  theorem  that, 
if  the  determinant  of  a  system  of  n  homogeneous  equations  van- 
ishes, it  is  always  possible  to  satisfy  the  equations  through  values 
of  the  unknown  quantities  that  are  not  all  equal  to  zero. 

Instead  of  the  equation  [16]  we  subject  the  variables  to  the 
somewhat  more  general  equation 

</»(;i;„  X^, ^„)  =  1, 

where  »/»  denotes  a  homogeneous  function  of  the  second  degree, 
which  is  always  positivef  and  is  only  =  o  when  the  variables 
themselves  vanish. 

18.  If  we  form  the  system  of  equations 

[24]  ^^—e^^  =  o     (X=l,  2,....«), 

then  these  equations  may  always  be  solved,  if  their  determinant 
vanishes. 

This  determinant  is  exactly  the  same  as  that  in  [23]  for  the 
case  that 


^=2- 


2 

X' 
X=l 


We  assume  that  e=^k-\-li,  and  that  we  have  found 

as  a  system  of  values  that  satisfy  the  equations  [24]. 
We  must  consequently  have 

■  •L+'nni)=o.  (\=i,  2,..../^) 

Since  the  real  and  the  imaginary  parts  of  these  equations  must  of 
themselves  be  zero,  it  follows,  when  we  observe  that  ^x  and  y^\ 
are  linear  functions  of  the  variables,  that : 

«^x(^i,  ^2,. .  .  -D—k^k  (^1,  ^2,-  •  •  -D  +  l^x  (Vv  Vz,---  •Vn)^o, 


*  Weierstrass,  Berlin.  Monatsbericht.  1868.  May  18  ; 
Cf.  also  Kronecker,  Berlin.  Monatsbericht.  1874,  p.  1 

t  That  we  are  justified  in  making  this  assumption,  is  seen  from  the  lemma,  which 
for  the  sake  of  continuity  in  the  argument  is  not  inserted  until  the  end  of  the  present  dis- 
cussion'(arts.  21,  22  and  23). 


of  Functions  of  Several  Variables. 


67 


19.  Multiply  these  equations  respectively  by  tjx  and  ^x,  take 
the  summation  over  them  from  1  to  n,  and  subtracting  one  of  the 
resulting  equations  from  the  other,  then,  since 


■n\ 


VX 


\ 


we  have 


or, 

[25] 


X  X 


If  it  is  possible  to  find  systems  of  values  of  the  quantities  x^, 

X2, x„  which  satisfy  the  equation    [24]   under  the  assumption 

that  e=^  +  li,  then  these  values  must  satisfy  at  the  same  time 
[25]  ;  but  since  after  our  hypothesis  the  quantity  within  the 
brackets  cannot  vanish,  it  follows  that  /  must  be  equal  to  zero ; 
that  is,  every  value  of  e  for  which  the  determinant  vanishes, 
is  real. 

Hence  we  have  the  theorem  : 

/n  order  that  a  quadratic  form  (j)  (x^,  x^,  ."■  ■ .  x„)  be  contin- 
uously positive,  it  is  necessary  and  sufficient  that  the  develop- 
ment of  the  determinant  [23]  which  admits  of  only  real  roots, 
when  expanded  in  powers  of  e,  viz. — 


[26] 


-B^  g"-^  +  i?2  e"-^'— .  . .  .  +(— 1)''^„=  o. 


consist  ofn-^1  terms  and  that  these  term-s  be  alternately  posi- 
tive and  negative. 

If  the  function  is  to  be  continuously  negative,  then  the 
equation  [26]  must  be  complete  and  have  continuation  of  sign. 

20.  Lemma.  If  a  homogeneous  function  of  the  second  de- 
gree «/»  {xi,  x^. . .  .x„)  can  become  zero  for  any  system  of  real  values 
of  the  variables  which  are  not  all  zero,  then  \}j  may  be  both  posi- 
tive and  negative,  it  being  presupposed  that  the  determinant  of  t|» 
is  different  from  zero. 


68  Theory  of  Maxima  and  Minima 

Let  the  function  ^  vanish  for  the  system  of  values  (^i,  ^2' ^n) 

and  instead  of  x-^,  x-^, x„  write  in  i/>  the  arguments  $i  +  Ci  ^,  ^2+ 

C2  k, f„  +  c„  k,  where  the  c's  are  indeterminate  constants. 

Developing  with  respect  to  powers  of  k  we  have 


a=n 


»/'(^i+  c^k,  ^2+  c^k, f„  +  <:„  k)^2k  2  ^^  V*"  (fi.  ^2. D 


1=1 


+  k^y\i  {c^,Cj, c^) (i). 

By  hypothesis  the  f 's  are  not  all  zero,  and  the  determinant  of  v|/ 

being  different  from  zero,  it  follows  that  T/»a  (a=l,  2, n)  can 

not  all  be  zero,  since  otherwise  the  equations  i/»a  =  <?  could  be  satis- 
fied by  the  ^'s  in  question,  and  consequently  the  determinant 
would  vanish,  which  is  contrary  to  our  supposition.  Since 
further  Ca  (a=l,  2, n)  are  arbitrary  constants,  we  may  so 


0=»i 


choose  them  that  ^Co  t/»o  (^1,  I2, |^„)  is  not  equal  to  zero. 

a=l 

Now  by  taking  k  sufficiently  small  we  may  cause  the  sign  of 
the  expression  {i)  to  depend  only  upon  the  first  term  on  the  right- 
hand  side  of  that  expression. 

Hence  if  we  choose  k  positive  or  negative,  we  have  systems 
of  values  {x^,  X2, .  . .  .x^)  which  make  ^  positive  or  negative. 

21.  The  determinant  of  the  system  of  equations  [24]  is 
formed  from  the  partial  derivatives  of 

<l>(xi,  X2, x^)  —  e^(  Xi,  X2, x„), 

that  is,  from 

<f)a  (X^,  X2, X„) ei//a(^i,  X2, Xn)=^0, (u) 

(a=l,  2, n) 

where  6a  and  \ba  denote  -z-^~   and  ^r-;^,  respectively.     If  this 

2   OXa  2   OXa 

determinant  is  equal  to  zero  for  a  value  of  e,  this  means :    we  can 

give  to  the  variables  x^,  X2, x^  values  that  are  not  all  zero  and 

in  such  a  way  that  the  n  equation  (n)  exist.  Let  this  value  of  e 
be  e=k  +  li;  then  if  />(?,  we  shall  show  that  the  function  xjf  can 
have  both  positive  and  negative  values. 


of  Functions  of  Several  Variables. 


69 


Denote  the  system  of  values  (;Vi,  x-^,...  .,-tr„)  which  satisf}'  the 
equation  (m)  by 

Xa.   =  ^a  +   i  r)a        (a=^  1,  2,  ....  w)  ; 

then,  as  in  art.  19,  it  may  be  proved  that 

Since  by  hypothesis  /is  not  zero,  the  equation  (ui)  can  only  exist 
either,  when  »/)  (^i,  ^3, .  .  .  .  f„)  and  i/»  (tJj,  tjj-  •  •  •  •  "^n)  have  opposite  val- 
ues, and  then  it  is  proved,  what  we  wish  to  show,  that  i/(  can  have 
both  positive  and  negative  values ;  or,  when  the  two  values  of  the 
function  are  both  zero,  and  then  from  what  was  seen  in  the  pre- 
ceding article,  «/»  can  take  both  positive  and  negative  values. 

22.     In  the  same  connection  it  is  intere.sting  to  prove  the  fol- 
lowing theorem  :     If  the  determinant  formed  from  the  partial 
derivatives  of  the  homogeneous  quadratic  form  ^  {x^,  x^,...  .x„) 
is  different  from^  zero,  and  if  am^ong  the  infinite  number  of 
quadratic  forms  : 

>i<i>(Xi,  X2, ^„)  +  /^  ^(^1,  ^2. ^n). 

there  is  one  definite  quadratic  form.,  the  determinant  formed 
from  the  partial  derivatives  of 

<l> (x^,  X2, X,,) — e  xjj  {xi,  X2, x^) 

vanishes  for  only  real  values  of  e. 

The  theorem  will  also  be  true,  if  the  determinant  of  ^  (and 
not  as  assumed  of  i/»)  is  different  from  zero. 

Let  Xj  <^  +  /Ai  i//  be  a  definite  quadratic  form,  and  write 

We  shall  further  choose  two  constants  \  and  /Aq  in  such  a  way 
that,  when  we  put 

<^  is  different  from  zero. 

We  know  from  the  previous  article  that  the  determinant  formed 

from  the  equations 

^a — k^^^o     (a=l,  2, w) 


70  Theory  of  Maxima  and  Minima 

can  only  vanish  for  real  values  of  k. 
The  equations 

<^a  — k xjjci.  =  o     (a=l,  2,....n) (iv) 

may  be  written,  in  the  form 

(X,, —  k  \i)  <f)a  +  {fio — k  ix,^)  xpa.  =  0     (a  =1,2, n), 

(a=l,2, n) 

If  we  eliminate  {x^,  X2, x„)  from  these  equations,  we  must 

have  the  same  determinant  for  their  solution  as  from  the  equa- 
tions (iv). 

Hence  every  k  which  causes  this  last  determinant  to  vanish, 
must  also  cause  the  first  determinant  to  vanish.  But  the  ^'s  are 
all  real.     Hence,  if- we  form  from  them  the  n  expressions 

^^k  1^1— lip 
\ — k  Xj 

these  quantities  must  also  be  real. 

Hence  the  determinant  of  the  n  equations 

^a — e^a  =  o     (a=l,  2, n) 

has  always  n  real  roots  e. 

We  may  therefore  say  :  If  a-inong  all  the  quadratic  forms 
which  are  contained  in  the  form, 

\<f>{x,,X2, X„)    +  flxl){Xi,X2, x„), 

there  is  one  which  can  have  only  positive^  or  only  negative  val- 
ues, then  the  determinant  of(f> — e  \jj  will  have  only  real  roots., 
it  being  assumed  that  the  determinant  of(f>,  or  of\jj,  is  not  zero. 

The  theorem  in  art.  17  is  accordingly  proved  in  its  greatest 
generality. 

23.  The  cavse  where  equations  of  condition  are  present  may 
be  easily  reduced  to  the  case  already  considered.  The  determin- 
ant [20]  was  the  result  of  eliminating  the  quantities  x^-,  x-^,.  .  .  .x, 
Ci,  ^2,  ■  •  •  •  ^m  from  the  n-\-  m  equations  : 


n. 


of  Functions  of  Several  Variables. 


71 


y-^n 


P=m 


[18]         2^^/^  x^,—  e  XX  -V  2  ^p  ^p^  ^°      ^^=^'  2'-  •  •  •^^' 

p=i 


M=l 


[15]         ^p=2«.^^M=^     (p=l,  2,'....^). 


M=i 


Since  the  result  of  the  elimination  is  independent  of  the  way  in 
which  it  has  been  effected,  we  may  first  consider  m  of  the  quanti- 
ties X,  say  :  x-^,  x^^. .  . . x„,  expressed  by  means  of  the  equations 
[15]  in  terms  of  the  remaining  n — m  of  the  x's,  which  may  be  de- 
noted by  ^1,  l^j, f„_„,.     We  thus  have 


[27] 


X^ 


2  O^^^     (/^  =  l.  2, m). 


v=l 


Through   the    substitution    of    these    values,  let    <^  (^i,  x^, . 
.  .x„)  be  transformed  into  <f)  (f„  fj-  •  •  •  -l^ii-m)  and  the  equation 


2   ^X  =^1.    into    xj/($u^2^ ^n-m)=   1  . 

X=l 

The  function  \j}  is  continuously  positive,  and  is  only  equal  to  zero, 
when  the  variables  themselves  all  vanish. 

The  equations  [18]  may  be  written  in  the  form : 

i9</>       ...  ,  ^     dd. 


2d  XX 


—  exx+  2 


p=i 


e^  -^-1^=0     (X-1,  2 n). 

d  XX 


Multiply  these  equations  respectively  by 
and  adding  the  results,  then  since 

^  d  6p    d  x\  _d  Op 


d  x\ 


(X=l,  2 n), 


d  x\    d^,       d  ^v' 


x=« 


•^  9  <^     dx\      d  (}) 


\^ 


d  x\    d$,       d  ^/ 


72  Theory  of  Maxima  and  Minima 

'K=n 

-2  -^ 


2dx\ 1^ 

x=i  x=i 


we  have  the  following  equations: 


1    d<l>        1,9^  ^2,   ^^   9^p 


^wr2^x+2^^-a?7=^- 


2    36  ~2       36  ^     X  '   ^  "  af, 

(v=l,2,....n) 

The  last  term  of  this  equation  drops  out,  if  we  substitute  in  it  the 
expressions  [27],  since  the  6p  expressed  in  the  ^'s  vanish  identic- 
ally, and  we  have  the  equations 

[28]  ll—e^=o     {v=l,2,....n-m). 

3  6         o  & 

Now  give  to  the  v  all  values  from  1  to  n — m,  and  we  have  a  sys- 
tem of  n — m  linear  homogeneous  equations,  from  which   we  may 

eliminate  the  yet  remaining  6,  ^2» 6-m  •      '^he   result  of  this 

elimination,  which  is  an  equation  in  e,  must  agree  with  [20].  The 
equations  [28]  are,  however,  created  in  exactly  the  same  manner 
as  the  equations  [24].  If  then  A  g  is  the  determinant  of  these 
equations,  it  follows  that  the  roots  of  the  equation  A  e=  o  are  all 
real. 

24.  As  the  solution  of  the  theorem  proposed  in  art.  13,  the 
final  result  is : 

/n  order  that  the  homogeneous  function  of  the  second- 
degree 

<f>ixi,  X2, x„)=^Ax^xx  Xfj,, 

X,  M 

de  continuously  positive  for  all  systems  of  values  of  the  quan- 
tities Xi,  X2, . . . .  X,,,  which  satisfy  the  m  linear  homogeneous 
equations  of  condition, 

^P^2  ^PM^M^^   (/)=l,2,....w). 


of  Functions  of  Several  Variables. 


73 


it  is  necessary  and  sufficient  that  the  form  of  the  equation  [20], 
developed  with  respect  to  powers  of  e  and  which  has  only  real 
roots,  consist  of  n — w  +  1  terms  and  that  the  signs  associated 
with  these  terms  be  alternately  positive  and  negative.  There 
m^ust,  however,  be  only  a  continuation  of  sign,  if  <}>  is  to  be 
continuously  negative. 

The  above  method  was  first  discovered  by  Lagrange,  who 
did  not,  however^  sufficiently  emphasize  the  reality  of  the  roots 
of  equation  [20]. 


APPLICATION  OF  THE   CRITERIA    JUST   FOUND   TO   THE   PROBLEM 

OF   THIS   CHAPTER. 

25.  We  have  determined  the  exact  conditions  that  a  homo- 
geneous quadratic  form  be  definite  for  the  case  where  the  vari- 
ables are  to  satisfy  equations  of  conditions,  and  in  a  manner 
entirely  symmetric  in  the  coefficients  of  the  given  function  together 
with  those  of  the  given  equations  of  condition. 

At  the  same  time  with  the  solution  of  this  problem,  the 
problem  of  maxima  and  minima  which  we  have  proposed  in  this 
chapter,  is  solved. 

26.  Having  regard  to  the  remarks  made  in  art.  7  and  art.  10 
we  have  as  a  final  result  of  our  investigations  the  following 
theorem. 

Theorem.  1/  those  positions  are  to  be  found  on  which  a 
given  regular  function  T  (x^,  X2, . . . .  x^)  has  a  m-axim.um-  or 
minim^um  value  under  the  condition  that  the  n  variables 
x-^,  X2,....  x„  satisfy  the  m,  equations 

[aj  f^{x^,X2 x^  =  o     (X=l,  2, m), 

where  f  are  likewise  regular  functions,  we  write 


P=ra 


\b-\ 


^+  2  ^p  /p=   G {x^,  Xj, x„), 


p-i 


and  seek  the  system  of  real  values 

•^1)   -''2>  •  •  •  •  ■^nt     ^l!    621  •  ■  •  •  e„  , 

which  satisfy  the  n^m  equations 


74 


Lc] 


Theory  of  Mayiima  and  Mmtma 
-.0     (X  =  l,  2, n), 


fy^  =  o    (X=.l,  2 m). 

I/{ai,  a^,...  .a„)  is  such  a  system  of  values  of  x^,  X2,...  .x„,  then 
we  develop  the  difference 

G  {a^-^h^,  «2  +  >^2' «n  +  '^,.) —  G  («!,  a2, a„) 

with  respect  to  powers  of  h,  and  have  {since  no  terms  of  the 
first  dimension  can  appear,  owing  to  equations  \c\)  the  follow- 
ing development: 

[^J        G{a^^h^,  a^Arh^, a„  +  ^„)  —  G{a^,a2, a 


11/ 


=  ^2    ^M-(^l''^2' ^n)    ^M     ^0  + 


M,  " 


We  must  next  see  whether  the  function 

M  ^(KK....h:).^^G^,h^  K 


M,  •' 


is  continuously  positive  or  continuously  negative  for  all  systems 
of  values  of  the  h's  which  satisfy  the  m  equations 


li^7t 


[/]  ^/p^iauav-  -a.)  h^^o     (p=l,2, .  . .  .m). 


M-i 


To  do  this  we  form  the  determinant 


lg\ 


^U        ^'         L7-12, U^jn,          /ii,  /21,  .  -frnx 

Gzx            ,         Cr22        e,    .  .  Cr2„,         /i2)  7221  •  •  /  m2 

^nl           >         ^n2i ^nn~^iyiii>y2n'  •  -/mn 

/n             ,      /l2. /in  1            0,    O  ,.  .  O 

/ml             .       /m2 /mm              C*  ,     0   ,..   0 

aw^  this  determinant  put  =^o  is  an  equation  of  the  m — n  degree 
in  e,  which  has  only  real  roots.  Developing  the  determinant 
with  respect  to  powers  of  e,  we  have  to  see  whether  the  develop- 
tnent  consists  of  n — m^\  terms  with  alternately  positive  and 
negative  sign  or  with  only  continuation  of  sign. 


of  Functions  of  Several  Variables. 


75 


If  the  first  is  the  case,  the  function  ^  is  continuously  posi- 
tive and  the  function  F  has  on  the  position  (ci,  a^,. . . . a„)  a 
MINIMUM  value,  if  on  the  contrary  the  latter  is  true,  then  <i> 
is  continuously  negative  and  F  has  on  the  position  {a^,  aj^.-a^) 
a  MAXIMUM  value. 

This  criterion  fails,  however,  when  <^  vanishes  identically, 
because  the  quantities  Gt^  v  vanish  for  the  position  (oti,  «2.  •  •  •  •  ^n)  5 
and  it  also  fails  when  the  smallest  or  greatest  root  of  A  e  =  o  is 
zero,  since  in  this  case  we  may  always  so  choose  the  h'%  that  <^ 
vanishes  without  the  A's  being  all  identically  zero. 

In  both  cases  the  development  [^/]  begins  with  terms  of  the 
third  or  higher  dimensions,  and  for  the  same  reason  as  that  stated 
at  the  end  of  the  last  chapter  we  may  assert  that  in  general  no 
maximum  or  minimum  will  enter  on  the  position  (ati,  a-i,...  .a„). 

We  give  next  two  geometrical  examples  illustrating  the  above 
principles. 

27.  Problem  I.  Determine  the  greatest  and  the  smallest 
curvature  at  a  regular  point  of  a  surface. 

If  at  a  regular  point  /^  of  a  plane 
curve  we  draw  a  tangent  and  from  a  neigh- 
boring point  P'  on  the  curve  we  drop  a 
perpendicular  P' Q  upon  this  tangent,  then 
P'Q  _i^^ 


the  value  that  2 


A5 


approaches. 


. 

^^^^ 

^^Cr**^''* 

p 

M 

&Y       1 

V 

/ 

J 

PP' 

if  we  let  P'  come  indefinitely  near  P,  is 
called  the  curvature  of  the  curve  at  the 
point  P.       If  the  curve  is  a  circle  with 

radius  r,  then  the  above  ratio  approaches  — 

as  a  limiting  value,  and  is  therefore  the  same  for  all  points  of  the 
circle.  Now  construct  the  osculating  circle  which  passes  through 
the  two  neighboring  points  P  and  P'  of  the  given  curve.  The  arc 
of  the  circle  PP'  may  be  put  equal  to  the  arc  PP'  of  the  curve, 
when  P  and  P'  are  taken  very  near  each  other,  and  consequently, 
if  r  is  the  radius  of  this  circle,  the  curvature  of  the  curve  is  de- 
termined through  the  formula 

2P'Q  _  1 


76  Theory  of  Maxima  and  Minima 

The  quantity  r  is  called  the  radius  of  curvature  and  the 
center  M  of  the  circle  which  lies  on  the  normal  drawn  to  the 
curve  at  the  point  P  is  known  as  the  center  of  curvature  at  the 
point  P.  The  curvature  is  counted  positive  or  negative  accord- 
ing as  the  line  P' Q,  or  what  amounts  to  the  same  thing,  yl/'/'has 
the  same  or  opposite  direction  as  that  direction  of  the  normal 
which  has  been  chosen  positive. 

If  we  have  a  given  surface  and  if  the  normal  at  any  regular 
point  of  this  surface  is  drawn,  then  every  plane  drawn  through 
this  normal  will  cut  the  surface  in  a  curve,  which  has  at  the  point 
P  a  definite  tangent  and  a  definite  curvature  in  the  sense  given 
above. 

The  curvature  of  this  curve  at  the  point  P  is  called  the 
curvature  of  the  surface  at  the  point  P={^x,y,  z)  in  the  direc- 
tion of  the  tangent  which  is  determined  through  the  normal 
section  in  question. 

Following  the  definitions  given  above  it  is  easy  to  fix  the 
analytic  conception  of  the  curvature  of  a  surface  and  then  to 
formulate  the  problem  in  an  analytic  manner: 

If  P'  =  {^x' ,  y ,  z')  is  a  neighboring  point  of  P  on  the  surface, 
the  equation  of  the  surface  may  be  written  in  the  form: 

[2]  o=F,  (x'-x)  +  P\  if-y)  +  P\  (z'—z) 

+y  I  ^n(^'—^y+-F^{y—yy+p22i^'—'^y+2F,,{x'—x){r—y) 

+  2F^(y'~y)iz'-z)  +  2F,lz'-^){x'-x)  |  + , 


where  P\  =.  ,^—  ,    /Tj  =  -—  ,     F^ 


d  F      y^  _3  F       j^       d  F 
X  ay  a  z 


/r_?!^      zr_3!^      /r_^ 
^'^  -d^'    ^~  dy^'    ^^  dz^' 

„         d^F        J.         d^F         zr         ^'F 


dxdy  '  dydz  '  dzdx 

The  equation  of  the  tangential  plane  at  the  point  P  is 
[3]  F,  {$-x)  +  F,  (v-y)  +  F,  a~z)=o. 


of  Functions  of  Several  Variables.  77 

If  we  therefore  write  for  brevity 

and  take  as  the  positive  direction  of  the  normal  of  the  surface  at 
the  point  /"that  direction  for  which  //is  positive,  tlien  the  direc- 
tion-cosines of  this  normal  are 

^^  ^2  and    ^3. 
H     H  H 

Consequently  the  distance  from  P'  to  the  tangential  plane  is 


[5] 


PQ 


-^{x'-x)+  f  (y-r)+  §  (^'-^). 


The  positive  or  negative  sign  is  to  be  given  to  the  expression 
on  the  right-hand  side  according  as  the  length  P' Q  has  the  same 
or  opposite  direction  as  that  direction  of  the  normal  which  has 
been  chosen  positive. 

In  the  first  case,  paying  attention  to  [2],  which  has  to  be 
satisfied  since  P'  lies  upon  the  surface,  we  have 


[6] 


2P'Q 


PP' 


F,lx'-xf  +  FAy'-yf  +  FJ^ z'-zf  +  2F,l x'-x)  {y'-y)  + 


where  S^={x—xy-\r{y—yf^-{2'—z)\ 

In  the  case  where  P' Q  is  contrary  to  the  positive  direction 
of  the  normal,  we  must  give  the  negative  sign  to  the  right-hand 
side  of  [6]. 

Now  let  P'  approach  nearer  and  nearer  P,  then  the  quantities 

^ — X     y' — y 


S 


S 


z' — z 


which  represent  the  direction-co.sines  of  the  line  PP  become  the 

direction-cosines  of  the  tangent  at  the  point  /"of  the  normal  section 

that  is  determined  through  P' .     Representing  these  by  a,  /8,  y  and 

.     .  .                              P' O 
the  limiting  value  of  2 by  k,  then  is 


PP 


[7]    /c=-|,|   Fy,a^^F^^\F^f-\-'ZFr,o.^^2F^^y^2F,,yo.    |, 


78  Theory  of  Maxima  and  Minima 

In  this  formula  k  represents  the  curvature  of  the  surface  in  the 
direction  determined  by  a,  )8,  y.  This  is  to  be  taken  positive  or 
negative  according  as  the  direction  of  the  length  MP,  where  M  is 
the  center  of  curvature,  corresponds  to  the  positive  direction  or 
not. 

If  the  coordinates  of  the  center  of  curvature  are  represented 
by  Xq,  j/q)  ^0  ^•iid  the  radius  of  curvature  by  p,  then  is 

x—Xo=pjj  ; 

or,  since  k  = —  , 
P 


f     X      Xq 


Fr 


^11^2  +  ^22/8'+ +2F,,ya 


[8}  {  y-y^=p^^  ^.^^^  ^_^  ;.....  ^2/^31  yo. ' 

- F^ 

^n<^'+^22/8H +2/^31  r« 

Since  H  does  not  appear  in  these  expressions,  v^^e  see  that  the 
position  of  the  center  of  curvature  is  independent  of  the  choice  of 
the  direction  of  the  normal. 

Suppose  that  the  normal  plane  which  is  determined  through 
the  direction  a,  /S,  y  is  turned  about  the  normal  until  it  returns  to 
its  original  position.  Then  while  a,  /8,  y  vary  in  a  definite  man- 
ner, the  function  k  of  a,  /S,  y  assumes  different  values  at  every 
instance,  and  since  it  is  a  regular  function,  it  must  have  a  maxi- 
mum value  for  a  definite  system  of  values  (a,  ^,  y)  and  likewise  also 
a  minimum  value  for  another  definite  system  of  values  (a,  /8,  y). 

/TV  •  1 

The  quantity  —  has  the  same  value  for  all  normal  sections 

that  are  laid  through  the  same  normal.*  We  have,  therefore,  to 
seek  the  systems  of  values  (a,  ^S,  y)  for  which  the  expression 

^nct'  +  /^22/8'  +  ^337"  +  2^i2«i8  +  2F^^y  +  2i^3ir« 
assumes  its  greatest  and  its  smallest  value. 


*  See  Salmon,  A  Treatise  on  the  Apalytic  Geometry  of  Three  Dimensions.    Fourth 
Edition,  p.  259. 


of  Functions  of  Several  Variables. 


79 


We  have  also  to  observe  that  the  variables  a,  /8,  y  must  satisfy 
the  equations  of  conditions: 

,2       I         cP.      s      -Jl 


[9] 


I  a^   +    ^   +  y2  _i. 

the  first  of  which  says  that  the  direction  which  is  determined 
through  a,  /3,  y  is  to  lie  in  the  tangential  plane  of  the  surface  at 
the  point  P,  while  the  second  equation  is  the  well  known  relation 
among  the  directions-cosines  of  a  straight  line  in  space. 
Following  the  methods  indicated  in  art.  26  we  write 

[10]         G=h\a}^t\^^ +2/^3.  ya 

—  g(aH/3'  +  r'— 1)   +  26'(>^t«  +  ^2i8  +  i^3V), 

and  we  then  have  (art.  26,  [c])  to  form  the  equations: 


dG 


0, 


d  G 


o, 


dG 


o. 


[11] 


da       ^'   d  fi       "'    8y 

from  which  we  must  eliminate  a,  /3^  y  and  e'. 
These  equations  are : 

-e)  a+  F,2p         +  F^y 
+  (F22—e)p+  Fj^y 

+  F^  p         +  F^y 

where /^xm=/vx     (X, /x=1,  2,  3). 
Through  elimination  we  have 

•''  11        ^  )     -' "*  12  >     ■''  13 

•^21  .     ^22 e,    Ft2 

■^31  >     -'    32  »     -'    33 

This  is  an  equation  of  the  second  degree  in  e,  and  consequently 
gives  us  two  valu'^s  e^  and  e^,  which  are  maximum  and  minimum 
values,  since  both  maximum  and  minimum  values  enter,  as  shown 
above.  Multiply  the  three  first  equations  [11]  by  a,  yS,  y  respec- 
tively and  adding  the  results,  we  have 


[12] 


j^F^e  =0, 
^F^e  =0, 
+  (^33— e)y  +/^3e'  =£?, 

=  0, 


F, 

F^ 

F.     -'' 

o 


80  Theory  of  Maxima  and  Minima 

[13]       i^„  a'  +  /';2/8'  +  ^33 r'  +  2/^i2 «y8  +2/^23/3  v  ^2F^^ia=e. 
Hence,  from  [7]  we  have 

p    h' 

Consequently  the  two  principal  curvatures  at  the  point  P  have 
the  values 


[14] 


[15] 


•  1      e. 


2 


Pz     H 


and  the  coordinates  of  the  corresponding  centres  of  curvature  are 
found  from  the  formulae 


[16]  '' 


1\  h\  F. 

^1  ^1  ^1 

)  /'^i  F.  F, 

V  ^2  ^2  ^2 

In  order  to  determine  e,  let  us  write 

Ai=(^'22— ^)  {Fyi—e)  —  F^^ 
F>n=  F^^F^^—Fi^iF^  —  e), 

and  form  from  these  the  corresponding  quantities  through  the 
cyclic  interchange  of  the  indices.  Equation  [12]  may  be  written 
in  the  form* 

F>nF,'  +  D^F}  -V  D^F}  +  2D,,F,F^  +  2D^F^F,  +  2D,,F,F,=  o. 

Developing  this  expression  with  respect  to  powers  of  e  we  have 

[17]  H^e'  —  Le  +  M=o, 

where  Z  =  ^n^n  +  ^22  +  /^33)-(^u/^lH/^22/'V  +  /^33^3^) 

+  2Fn  F,  F^  +  2/^^23  F^  F,  +  2F,,  F,  F, 

and  ^=(/^22^33-/'V)/^l'+(/^33^n-^V)    F,' 

-{- {.-^  11  ■'' 22       -''12  ) -^3   +  (-''l2-''l3'      '■''73  ^11) -^2  ■'^3 

+  (Fzi F^i—F^, F22) F^ F^  +  (F^i F32—Fn F^^)  F,  Fj. 


*  See  Salmon,  p.  257. 


of  Functions  of  Several  Variables.  81 

From  [17]  we  have  at  once  the  values  of  the  sum  and  the 
product  of  the  two  principal  curvatures,  viz. — (see  equation  15): 

[18]  r    '^ 

J  1    ^  ^L. 

V  Pk92  ^' 

We  have  thus  expressed  the  sum  of  the  reciprocal  radii  of  curva- 
ture and  also  the  measure  of  curvature  of  the  surface  at  the  point 
P  directly  through  the  coordinates  of  this  point. 

Although  the  formulae  are  somewhat  complicated,  thej^  are 
used  extensively  and  with  great  advantage. 

In  the  case  of  -minimal  surfaces^,  which  are  characterized 
through  the  equation 

Pl  +  />2  =  0, 

we  have  L~o. 

This  is  therefore  the  general  differential  equation  for  mini- 
mal surfaces. 

Art.  28.  Problem  II.  From  a  given  point  {a,  b,  c)  to  a 
given  surface  F{x,y,z)^o,  draw  a  straight  line  whose  length 
is  a  maximum  or  a  minim-um. 

Write 

G^{x—ay+{y-by  +  {3-cy  +  2)^F{x,y,z) {t) 

Then  the  quantities  x,  r,  z,  X  are  to  be  determined  (see  [c]  art.  26) 
from  the  following  equations: 

X — a  +  XFi=o, 

y—6+\F,=  o,  ^ (^..^^ 

z—c  +  XF3=.  o, 
F  {x,  y,  z)  =  o. 

It  follows,  since  F-^,  /s,  F^  are  proportional  to  the  direction- 
cosines  of  the  normal  to  the  surface  at  the  point  {x,y,  z),  that  the 
points  determined  through  these  equations  are  such  that  lines 
joining  them  to  the  point  {a,  b,  c)  stand  normal  to  the  surface. 

If  P={x,  y,  z)  is  such  a  point,  then  to  determine  whether  for 
this  point  the  quantity 

{x-ay^{y-by+{z-cy 


See  papers  on  this  subject  in  the  first  numbers  of  the  Mathematical  Review. 


I 


82  Theory  of  Maxima  atid  Minima 

is  in  reality  a  maximum  or  a  minimum,  we  substitute  x-\rU,  y-\-v, 
z-\-w  instead  of  x,  y,  z  in  the  function  G.  The  quantities  u,  v,w 
are  of  course  taken  very  small. 

We  must  develop!  the  difference 

G{x+u,  y  +  v,  z  +  w)  —  G(x,  y,  2) {in) 

in  powers  of  u,  v  and  w. 

The  terms  of  the  first  dimension  drop  out  and  the  aggregate 
of  the  terms  of  the  second  dimension  is 

+  2/^12  ^z^-f  2/^23^^  +  2/^31  w  u) ....... .  .{iv) 

Since  the  point  {x^ti,  y^v,  s+zu)  must  also  lie  upon  the  surface, 
the  quantities  Uj  v,  w  must  satisfy  the  condition 

F-^UArF^V^F^W^^O, {v) 

where  the  terms  of  the  higher  dimensions  are  omitted  (See  [8]  of 
the  present  chapter). 

If  we  wish  to  determine  whether  the  function  »/»  is  continu- 
ously positive  or  continuously  negative  for  all  systems  of  values 
{u,  V,  vi)  which  satisfy  equation  (z^),  we  may  seek  the  minimum 
or  maximum  of  this  function  »/»  under  the  condition  that  the  vari- 
ables are  limited  besides  the  equation  {y)  to  the  further  restric- 
tion [cf.  (16)]  that 

'CP-^tP'^v? — 1=0 {vi) 

For  this  purpose  we  form  the  function 

«// — e^u^^iP'^vf- — \) -\-2^ {F-^u^ F^v ■{- F^ui) , . . .  .{vii) 

and  writing  its  partial  derivatives  with  respect  to  u,  v  and  w 
equal  to  zero,  we  derive  the  equations: 

/          e-l\                    e'             \ 
F2iU+\F^—  -^jv  +  F^i  w  +  —  F2=o,  >  {viu) 

F^iU  +  F^.v  +  IFs^—  ^j  w  +-^  F2=o. 

Eliminating  2^,  Z',  Zf, —— from  equations    [z^]   and  {vin)  we   have 

A 


of  Functions  of  Several   Variables. 


83 


here  exactly  the  same  system  of  equations  as  in  [12]  of  the  pre- 
ceding problem,  except  tha  there  ——and  e'  stand  in  the  place  of 

e  and  e'. 

Denote  the  two  roots  of  the  quadratic  equation  in  e,  which  is 
the  result  of  the  above  elimination,  by  gj  and  e-^  and  the  corre- 
sponding radii  of  curvature  of  the  normal  sections  by  pi  and  pj, 


then,  since 


e-\ 


has  the  same  meaning  as  e  in  the  previous  problem: 
X    '  H' 


Pi 
1 


e,-l 


Pi 


1^ 


where  the  positive  direction  of  the  normal  to  the   surface  is  so 
chosen  that  iY>-  o . 

If  for  the  position  (x,  y,  2)  a  minimum  of  the  distance  is  to 
enter,  then  both  values  of  the  e  must  be  positive  ,  if  a  maximum, 
then  ^1  and  e^  must  be  negative. 

It  is  easy  to  give  a  geometric  interpretation  of  the  result  just 
obtained :  Let  PN  be  the  positive  direction  of  the  normal  and 
A  =  (a,  d,  c).  Then  from  («)  it  follows  that  the  length  from  A 
to  P  has  the  same  or  opposite  direction  as  PN  according  as  X.  is 
negative  or  positive. 

Hence  from  (w) 

A  P=—X//. 

If  the  centres  of  curvature  corresponding  to  pi  and  p^  be  denoted 
by  Ml  and  Mj ,  then  is 

\//         A  P 


M,P  = 


M^P 


e—1 

\H 
e^ — 1 


A  P 


e,—l 


Hence  ei=     }        and  ^2    = 

■iyi\  i 

If  then  J/i  and  M^  lie  on  the 
same  side  of  P  and  if  A  lies  be- 
tween Ml  and  M^  as  in  figs.  1 
and  2,  then  the  e's  have  differ- 
ent signs  and  there  is  neither 
a  maximum  nor  a  minimum. 


M-,A 
M^P 


M, 

A        M^ 

P 

^ 

Fig.  1. 

^ 

P 

Ml       A 

M, 

-> 

Fig.  2. 


M, 

*M^       P 

A 

Fig.  3. 

^ 

A 

M^       M^ 

P 

^^ 

Fig.  4. 

M, 

P        A 

M, 

-^> 

84  Theory  of  Maxima  and  Minima 

If  J/i   and  M^  lie   on  the 
same  side  of  P  and  A  without 

the  interval  M^ Mt_,  then  a 

maximum  will  enter  according 
as  A  starting  from  one  of  the 
centres  of  curvature  lies  upon 
the  same  side  as  P  or  not  (see 
figs.  3  and  4).  '^^-^ 

If  the  points  M^  and  Mj,  lie  on  different  sides  of  P  and  if  A 
is  situated  within  the  interval  M^.  . .  .Mj  as  in  Fig.  5,  then  there 
is  always  a  minimum.  If^  however,  A  lies  without  the  interval 
Ml ....  yJ/j,  then  there  is  neither  a  maximum  nor  a  minimum. 

In  whatever  manner  M-^  and  M2  may  lie,  if  A  coincides  with 
one  of  these  points,  then  one  of  the  tv^^o  values  of  e  is  equal  to 
zero  and  then  the  general  result  stated  at  the  end  of  art.  26  is 
applicable. 

The  case  may  also  happen  here  [see  art.  8  of  this  chapter] 
that  in  the  solution  of  the  equations  {ii)  and  {in)  a  singular  point 
of  the  surface  is  found  as  the  point  P,  at  which  Fi~o=Fi_^F^. 

For  such  a  case  we  cannot  proceed  as  above;  since,  there 
being  no  definite  normal  of  the  surface  at  such  a  pointy  the  deter- 
mination whether  for  this  point  a  maximum  or  minimum  really 
exists,  cannot  be  decided. 

The  general  remark  of  art.  8  indicates  how  we  are  to  proceed. 


of  Functions  of  Several  Variables. 


85 


CHAPTE)R  IV. 

SPECIAL   CASES. 

THE    PRACTICAL    APPLICATION    OF    THE  CRITERIA    THAT    HAVE 

BEEN    HITHERTO    GIVEN    AND    A    METHOD   POUNDED   UPON 

THE  THEORY  OF  FUNCTIONS,  WHICH  OFTEN  RENDERS 

UNNECESSARY  THESE  CRITERIA. 

1.  The  practical  application  of  the  established  criteria  is  in 
many  cases  connected  with  very  great,  if  not  insurmountable 
difficulties,  which,  however,  cannot  be  disregarded  in  the  theory. 
For  often  the  solutions  of  the  equations  [c],  art.  26,  of  the  previ- 
ous chapter  cannot  be  effected  without  great  difficulty,  if  at  all, 
and  therefore  also  the  formation  of  the  function  <^  is  impossible. 
It  also  happens,  even  if  the  function  ^  can  be  formed,  that  the 
discussion  regarding  the  coefficients  of  Ae=(?  is  attended  with 
much  difficulty.  Moreover,  the  formation  of  the  function  <^  and 
the  investigation  relative  to  the  coefficients  of  A  e  are  very  often 
unnecessary,  since  through  direct  observation  we  may  in  many 
cases  determine  whether  a  maximum  or  a  minimum  really  exists. 
If  it  happens  that  the  equations  [c]  admit  of  only  one  real  solu- 
tion {^i.e.  of  a  real  system  of  values  x^,  x^,. .  ■  .^„),  then  we  may  be 
sure  that  this  is  in  reality  the  maximum  or  minimum  of  the  func- 
tion. In  the  same  way,  if  we  can  convince  ourselves  a  priori 
that  a  maximum  and  a  minimum  exist,  and  if  it  happens  that  the 
equations  [c]  offer  only  two  real  systems  of  values,  it  is  evident 
that  the  one  system  must  correspond  to  the  maximum  of  the 
function,  the  other  system  to  the  minimum  value. 

The  determination  which  of  the  two  systems  of  values  gives 
the  one  or  the  other,  is  in  most  cases  easily  determined. 

2.  As  has  already  been  indicated  in  the  introduction  (see 
art.  1,  Chapt.  I.),  one  cannot  be  too  careful  in  the  investigation 
whether  on  a  position  which  has  been  determined  from  the  equa- 
tions [a]  and  [c]  of  art.  26,  Chapt.  III.  there  really  is  a  maximum 


^        OF  THE 

UN'iVERSlTY 

OF 


86  Theory  of  Maxima  and  Minima 

or  minimum,  since  there  are  cases  in  whicji  one  may  convince 
himself  of  the  existence  of  a  maximum  or  minimum,  when  in  reality 
there  is  no  maximum  or  minimum. 

For  example,  to  establish  Euclid's  theorem  respecting  parallel 
lines,  one  tries  to  prove  the  theorem  regarding  the  sum  of  the 
angles  of  a  triangle  without  the  help  of  the  theorem  of  the 
parallel  lines.  Legendre  was  able  indeed  to  show  that  this  sum 
could  not  be  greater  than  two  right  angles;  however,  he  did  not 
show  that  they  could  not  be  less  than  two  right  angles.  The 
method  of  reasoning  employed  at  that  time  was  as  follows:  If  in 
a  triangle  the  sum  of  the  three  angles  cannot  be  greater  than  180°, 
then  there  must  be  a  triangle,  for  which  the  maximum  of  the  sum 
of  these  angles  is  really  reached.  Assuming  this  to  be  correct,  it 
may  be  shown  that  in  this  triangle  the  sum  of  the  angles  =180°, 
and  from  this,  it  may  be  proved  that  the  same  is  true  of  all 
triangles. 

We  see  at  once  that  a  fallacy  has  been  made.  For  if  we  ap- 
ply the  same  conclusions  to  the  spherical  triangles,  in  the  case  of 
which  the  sum  of  the  angles  cannot  be  smaller  then  180°,  we  would 
find  that  in  every  spherical  triangle  the  sum  of  the  angles  is 
=  180°,  which  is  not  true. 

The  fallacy  consists  in  the  assumption  of  the  existence  of  a 
maximum  or  minimum;  it  is  not  always  necessary  that  an  upper 
or  lower  limit  be  reached,  even  if  one  can  come  just  as  near  to  it  as 
is  wished. 

On  this  account  also  in  our  case  the  assumption  of  the  exis- 
tence of  a  real  maximum  is  not  allowed  without  further  proof. 
We  therefore  endeavor  to  give  the  existence-proof.  For  this  pur- 
pose we  must  recall  several  theorems  in  the  theory  of  functions*. 

3.  We  call  the  collectivity  of  all  systems  of  values,  which  n 
variable  quantities  x-^,  X2,....x„  can  assume^  the  reahn  (Gebiet) 
of  these  quantities,  and  each  single  system  of  values  a  position  in 
this  realm.  If  these  quantities  are  variables  without  restriction, 
so  that  each  of  them  can  go  from  — oo  to  +  oo ,  we  call  the  realm 
considered  as  a  whole  (Gesamtgebiet)  an  n-ple  multiplicity  {n- 
fache  Mannigfaltigkeit).  For  example,  the  points  of  a  straight 
line  in  the  plane  form  a  simple  (einfach)  multiplicity.     The  points 


'Cf.  Chapt.  I.,  especially  arts,  16  et  seq. 


of  Functions  of  Several  Variables. 


87 


in  a  plane,  a  double  njultiplicity;  in  space,  a  triple  multiplicity.  All 
straio^ht  lines  in  space  form  a  \-pie  multiplicity.  For  consider 
two  planes,  then  in  each  of  them  a  point  is  determined  by  two 
quantities,  and  a  straight  line  by  two  points,  i.  e.,  by  four  quan- 
tities. Also  all  surfaces  of  spheres  in  space  form  a  \~ple  multi- 
plicity, since  every  sphere  needs  for  its  complete  determination 
the  three  coordinates  of  the  centre  and  the  radius  of  the  sphere. 

If  the  quantities  are  connected  with  one  another  by  equations 
of  condition,  so  that  m  of  them  (say)  may  be  considered  as  inde- 
pendent, then  we  call  the  collectivity  of  all  positions  an  m-ple 
multiplicity,  or  a  structure*  of  the  /wth  kind  in  the  region  of  these 
n  variables. 

For  example,  a  straight  line  in  space  is  a  structure  of  the 
first  kind  in  the  region  of  three  variables.  If  x^,  X2,....x„  are  in- 
dependent of  one  another,  then  we  say  a  definite  position  (aj,  a2,  ■ . 
.  .a„)  lies  on  the  interior  of  the  realm,  if  these  positions  and  al- 
so all  their  neighboring  positions  belong  to  this  region ;  it  lies  up- 
on the  boundary  of  the  realm,  if  in  each  neighborhood  as  small 
as  we  wish  of  this  position,  there  are  present  positions  which  be- 
long to  the  realm,  and  also  those  that  do  not  belong  to  it ;  it  lies 
finally  without  the  defined  realm,  if  in  no  neighborhood  as  small 
as  we  wish  of  this  position  there  are  positions  which  belong  to 
the  defined  region. 

If  the  quantities  x-^,  X2,....x„  are  subjected  to  nt  equations  of 
condition,  then  we  may  express  these  in  terms  of  n — m  independ- 
ent variables  Ui,  u-^, ....  ^„_m.  and  the  same  definition  may  be  ap- 
plied to  these  variables. 

4.  The  following  theorems  are  proved  in  the  theory  of  func- 
tions :  (1)+  If  a  continuous  variable  quantity  is  defined  in  any 
manner,  this  quantity  has  an  upper  and  a  lower  limit;  that  is, 
there  is  a  definitely  determined  quantity  g  of  such  a  kind  that  no 
value  of  the  variable  can  be  greater  than  g,  although  there  is  a 
value  of  the  variable,  which  can  come  as  near  to  g  as  we  wish. 
In  the  same  way  there  is  a  quite  determined  quantity  k  of  such  a 
nature  that  no  value  of  the  variable  is  less  than  k,  although  there 
is  a  value  of  the  variable  that  comes  as  near  to  k  as  we  wish. 


*See  Chapt.  I.,  art.  7  and  art.  9.     See  also  Chapt.  I.,  art.  16. 

tDini,  Theorie  der  Functionen,  p.  68.      See  also  a  paper  by   Stolz,   B.   Bolzano's 
Bedeutung  in  der  Geschichte  der  Infinitesimal  Rechnung.  Math.  Ann.,  bd.  XVIII. 


88  Theory  of  Maxima  and  Minima 

(2)*  In  the  region  of  n  variables  x^,  X2, .  . .  .  x^,  suppose  we 
have  an  infinite  number  of  positions  defined  in  any  manner, — let 
these  be  denoted  by  (yx[,  x^., .  .  .  .x'„), — further  suppose  that  among 
the  positions  we  have  such  positions,  that  x'„  can  come  as  near  to 
a  fixed  limit  a„  as  we  wish.  Then  we  have  in  the  region  of  the 
quantities  x^,  x^, . . .  .x^  always  at  least  one  definite  position  (^i,  ^2, 
. . .  .a„)  of  such  a  nature  that  among  the  definite  positions  (x[,  x'^, 
....x'„)  there  are  always  present  positions  that  lie  as  near  this 
position  as  we  wish,  so  that  therefore,  if  8  denotes  a  quantity 
arbitrarily  small, 

I  ^\— «x  I  <S(X=1,  2, n). 

This  position  lies  either  within  or  upon  the  boundary  of  the  de- 
fined region  {x[,  x'^,...  .x'^. 

5.  This  presupposed,  let  us  consider  a  continuous  function 
F{^x^,  X2, .  .  .  . Xn),  and  let  the  realm  of  the  quantities  x^^,  X2,  ■ . .  .x„  be 
a  limited  one,  so  that  therefore,  we  have  systems  of  values,  which 
do  not  belong  to  it.  If  for  every  possible  system  of  values  (x^,  x^, 
....x„),  we  associate  the  corresponding  value  of  the  function, 
which  may  be  denoted  by  x„+i,  then  we  have  defined  certain 
positions  in  the  region  oi  n  +  1  quantities.  For  the  quantity  x„+i 
there  is  according  to  the  first  theorem  an  upper  limit  a„+i;  con- 
sequently owing  to  the  second  theorem  there  must  be  within  the 
interior  or  upon  the  limits  of  the  defined  region  a  position  (a,,  a2, 
..a„,  a„+i)  of  such  a  nature  that  in  the  neighborhood  of  this 
position  there  certainly  exist  positions,  which  belong  to  the  re- 
gion in  question. 

The-.case  ^f  a  maxi77,um..  Now  if  it  Can  bc  showu  that  this 

position  lies  within  the  interior  of  the  re- 
gion, then  there  is  in  reality  a  maximum 
of  the  function  on  the  position  (a^,  a^, . . 
a„);  on  the  contrary,  if  the  position  lies 
on  the  boundry,  then  we  cannot  come 
to  a  conclusion  regarding  the  existence 
of  a  maximum  of  the  function  x^+i. 
It  may  in  many  cases  happen,  that 
/M  </(a)  >f(x.)  ^^^  ^^^  show,  if  {xi,  X2, .  . . .  x^)  is  any 


*Biermann,  Theo.  der  An.  Funk.,  p.  81;  Serret,  Diff.  et  Int.  Cal.,  p.  26. 


of  Functions  of  Several  Variables. 


89 


Th^    case   of  a   minimum. 


Cose  of  Q£>\/'npfof'C    approach 


UK)  <  fix)  <  /(X,) 

Fig.  3. 

Hittif"^^    on  tfit  JlmWiT^  poJltton   X,. 


fix,)  >flx)  >fix^. 


position  on  the  boundary  of  the  realm 
and  if  x„+-^  denotes  the  corresponding  val- 
ue of  the  function,  that  there  are  present 
within  the  realm  positions  for  which 
the  values  of  the  function  are  greater 
than  for  every  position  on  the  boundary. 
Then,  of  course,  the  position,  which  we 
are  considering  here,  cannot  lie  upon  the 
boundary,  and  it  is  clear  that  the  limit- 
ing value  of  the  function  can  be  assumed 
for  a  definite  position  within  the  inter- 
ior, since  the  function  varies  in  a  con- 
tinuous manner.  The  analog  is,  of 
course,  true  for  a  minimum.  If,  how- 
ever, it  does  not  admit  of  proof  that 
there  are  positions  on  the  interior  of 
the  defined  realm  for  which  the  value  of 
the  function  is  greater  or  smaller  than 
it  is  for  all  positions  on  the  boundary, 
then  nothing  can  be  concluded  regarding 
the  real  existence  of  a  maximum  or 
minimum;  the  position  {a-^,  a2, .  . .  .  a^) 
would  then  lie  on  the  boundary  of  the 
region,  and  there  might  be  an  asymptotic 
approach  to  the  limiting  value  a„+i 
without  this  value  being  in  reality 
reached  (cf.  Chapt.  I.,  art.  11).  This, 
however,  need  not  necessarily  be  the 
case. 

The  figures  give  a  plain  picture 
of  what  has  been  said  for  the  case 
y^/(^x),  where  x  is  limited  to  the  in- 
terval {Xi.  .X.  .X2). 


IT  IS  POvSSIBLE  THAT  THE  DIPPERENCE 

F{ai+hi,a2  +  k2,. .  .a„  +  h„) — F(ai,  a^, a„) 

IS  NEITHER  POvSlTlVE  NOR  NECzATIVE  BUT  ZERO  ON 
THE  POSITION  («!,  a-i,, (a!„) WHICH  IS  TO  BE  INVESTIGATED. 

6.     We  shall  now  consider  a  case  which  is  not  included  in 
the  previous  investigations  but  may  be  in  a  certain  measure  re- 


I 


90  Theory  of  Mayiima  and  Minima 

duced  to  them:  The  definition  of  the  maximum  or  minimum  of  a 
function  consists  in  the  fact  that  the  difference 

/^(ai+>^i,  a2+^2. a„+A„)— /^(ai,  ^2, a„) (0 

must  be  continuously  negative  or  continuously  positive.  There 
are  cases  where  a  maximum  or  a  minimum  does  not  appear  on  the 
position  («!,  ^2, . . .  .a„)  in  the  sense  that  the  above  difference  must 
be  positive  or  negative,  but  in  the  sense  that  the  difference  must 
be  zero. 

Suppose,  for  example,  we  have  the  problem:  Determine  a 
polygon  of  n  sides  with  a  given  constant  perimeter  6'  whose  area 
is  a  maximum, — a  problem  which  we  shall  later  discuss  more  fully 
(see  art.  10). 

If  this  maximum  is  attained  for  a  definite  polygon,  then  we 
may  at  pleasure  change  the  system  of  coordinates  by  sliding  the 
polygon  in  the  plane  without  altering  the  area. 

For  example,  let  /?=3,  and  (xi,j^i),  (^2.J^'2)  ^J^d  (x2,y2)  be  the 
coordinates  of  the  vertices  of  the  triangle.  Then  the  expression 
which  is  to  be  a  maximum  is 

where  the  variables  are  subjected  to  the  condition 


S^V{x^—x,y  +  {y2—}'ry+  ^{x^—x^y^iyr-yif 


+  v\x—x^y+{}'—yiy 


There  will  not  only  be  one  system  of  values  which  gives  for  F  a 
maximum  value,  but  an  infinite  number  of  such  positions;  since,  if 
we  take  the  triangle  in  a  definite  position,  we  may  move  it  in  its 
plane  at  pleasure.  This  is  therefore  a  case  where  the  difference 
(*')  is  not  positive  or  negative  but  zero. 

7.  Such  cases,  however,  may  be  reduced  to  maxima  and 
minima  proper,  if  we  choose  arbitrarily  some  of  the  variable  quan- 
tities. In  the  special  example  of  the  preceding  article  we  may 
assume  a  vertex  of  the  triangle  at  pleasure;  let  it  be  the  origin  of 
coordinates,  and  we  further  assume  that  one  of  the  sides  coincides 
with  the  positive  direction  of  the  ^-axis,   so  that  we  may  write 


of  Functions  of  Several  Variables.  91 

x-^=yy=yT:=o.  If  we  agree  that  the  triangle  is  to  lie  above  or 
below  the  ^-axis,  the  problem  is  completely  determinate. 

8.  In  so  far  as  the  necessary  conditions  for  the  existence  of 
a  maximum  or  minimum  are  concerned,  we  may  proceed  in  pre- 
cisely the  same  manner  as  we  have  hitherto  done;  since  under  the 
assumption  that  there  are  no  equations  of  condition,  we  have 

0=1 

If  a  minimum  is  to  be  present,  then  this  difference  can  never  be 
negative,  but  may  be  zero.     For  this  to  be  possible  the  first  de- 

rivatives  must  all  vanish.   Since,  if  the  sum  ^Aa/^o(«i,  ^jt  •  •  •  -(^^ 

had  (say)  a  positive  value  for  h^^^c-^,  hi=.CT_,. . .  .h^^c^,  then  we 
could  place  ha.  ^Cah,  and  then  choose  h  so  small  that  the  sign  of 
the  right-hand  side  of  {if)  depends  only  upon  the  sign  of  the  first 
term.  If  we  then  make  h  positive  or  negative  the  difference  would 
also  be  positive  or  negative. 

If  equations  of  condition  are  present,  it  may  be  showed  as 
above  that  the  derivatives  of  the  first  order  must  vanish,  since, 
if  all  these  derivatives  did  not  vanish,  we  might  express  some  of 
the  A's  through  the  remaining,  and  then  proceed  as  we  have  just 
done.  The  required  systems  of  values  {xx,  x^,-  ■  ■  ■  ^„)  will  there- 
fore be  determined  from  the  same  equations  as  before. 

9.  If  we  have  found  a  system  of  values  of  the  ;i;'s  which 
satisfy  the  equations  of  condition  of  the  problem,  then  in  the 
neighborhood  of  this  position  there  will  be  an  infinite  number  of 
other  positions  which  satisfy  the  equations.  These  last  are  char- 
acterized by  the  condition  that  the  difference  (*')  vanishes  identic- 
ally for  them. 

This  is  just  the  condition  that  made  impossible  the  former 
criteria,  by  means  of  which  we  could  decide  whether  a  maximum 
or  minimum  really  entered  on  a  position  («i,  fZz, .  . .  .a^  that  was 
determined  through  the  equations  in  Xi,  x^,....  x^. 


I 


92  Theory  of  Maxima  and  Minima 

One  must  therefore  seek  in  another  manner  to  convince  him- 
self which  case  is  the  one  in  question. 

This  is  further  discussed  in  the  following  example. 

10.  Problem.  Among  all  polygons  which  have  a  given 
number  of  sides  and  a  given  perimeter ,  find  the  one  which  con- 
tains the  greatest  surface-area. 

We  see  at  once  that  the  problem  proposed  here  is  of  a  some- 
what different  nature  than  the  problems  of  arts.  27  and  28  of  the 
previous  chapter,  since  the  existence  of  the  maximum  value  of  the 
function  is  no  longer  the  question  as  was  proposed  in  art.  4  of 
Chapt.  II.  and  held  as  fixed  throughout  the  general  discussions. 
For,  if  the  definition  of  the  maximum  is  such  that  the  function  on 
the  position  ((Xj,  a^,....a^  must  have  a  greater  value  on  this 
position  than  on  all  neighboring  positions,  then  in  this  sense  our 
polygon  could  certainly  not  have  a  maximum  area:  Since,  if  we 
had  such  a  polygon  on  any  position,  we  might  slide  the  polygon  at 
pleasure  without  changing  its  shape  and  consequently  its  area. 
Therefore,  only  a  maximum  of  the  area  can  enter  in  the  sense 
that  the  periphery  remaining  the  same,  an  increase  in  the  area  of 
the  surface  cannot  enter  for  an  infinitely  small  sliding  of  the  end- 
points.  We  consequently  cannot  apply  our  general  theory  with- 
out further  discussion. 

11.  Let  the  coordinates  of  the  n  end-points  taken  in  a  defin- 
ite order  be 

The  double  area  of  a  triangle,  which  has  the  origin  as  one  of  its 
vertices  and  the  coordinates  of  the  other  two  vertices  x^,  y-^  and 
Xi^,y-i  is,  neglecting  the  sign,  determined  through  the  expression 

x^y^  —  x^y^. 

In  order  to  determine  the  sign  of  this  expression  we  suppose 
that  the  fundamental  system  of  coordinates  is  brought  through 
turning  about  its  origin  into  such  a  position  that  the  positive 
X-axis  falls  together  with  the  length  01.  We  call  that  side  of  the 
line  01  positive  which  corresponds  to  the  positive  direction  of  the 
K-axis:  The  double  area  of  the  triangle  012  is  to  be  counted  posi- 
tive or  negative,  according  as  it  lies  on  the  positive  or  negative 
side  of  the  line  01. 


of  Functions  of  Several   Variables. 


93 


If  the  point  0  lias  the  coordinates  x^,,  y^,  the  double  area  of 
the  triangle  is 

2  Aoi2  =  (;t;i— ;ro)  (>'2— Jo)— (ji^i— To)  {x^—x^)  , 

where  the  above  criterion  with  reference  to  the  sign  is  to  be  ap- 
plied. » 

For  the  polygon  we  shall  take  a  definite  consecutive  arrange- 
ment of  the  points  (1,  2,....n)  and  besides  shall  assume  that  no 
two  of  the  sides  cross  each  other.  The  last  hypothesis  is  justifi- 
able, since  we  may  easily  convince  ourselves  that,  if  two  sides  cut 
each  other,  we  may  at  once  construct  a  polygon  whose  sides  do 
not  cut  one  another,  and  which  having  the  same  perimeter  as  the 
first  polygon  incloses  a  greater  area. 

Within  the  polygon  take  a  point  0  ^ 
(■^0.  j^'o)  and  draw  from  it  in  any  direction  a 
straight  line  to  infinity.  This  straight  line 
always  cuts  an  odd  number  of  sides  of  the 
polygon.  ' 

Now  if  we  follow  the  periphery  of  the 
polygon  in  the  fixed  direction  (1,  2,.  . .  .n) 
and  mark  the  intersection  of  a  side  by  the 
straight  line  with  -'-  1  or — 1,  according  as  we 
pass  from  the  negative  to  the  positive  side 
of  that  line  or  vice  versa,  then  the  sum  of  these  marks  is  either 
+ 1  or  — 1.  In  the  first  case  we  say  that  the  polygon  has  been  de- 
scribed in  the  positive  direction,  in  the  second  case  in  the  negative 
direction. 

It  may  be  proved*  that,  whatever  point  be  taken  as  the  point 
0  within  the  polygon^  and  in  whatever  direction  the  straight  line 
be  drawn,  we  always  have  the  same  characteristic  number  +1  or 
— 1,  if  in  each  case  the  positive  side  of  the  straight  line  has  been 
correctly  determined. 

12.     The  double  area  of  the  polygon  is 

2F^{x—x^)  {yz—y^—{x-r-x^)  (jJ'i— To)  +  {x^—x^)  {y^—y^) 

— (^3— ^o)  iy2—yo)+  ■  ■  ■  •+(x„—Xo)(yi—yo)—{x^—Xo)(yn—yo); 


or,       2F=x,  yr-x^y,  +  X2  yr-x^yz  +....+  x^y^-x^y^ 


(a) 


*  The  proof  is  found  in  Cremona,  Elementi  di  geometria  projetiva.    Rome.  1873. 


94  Theory  of  Maxima  and  Minima 

where  the  positive  or  negative  sign  is  to  be  taken  according  as 
the  polygon  has  been  described  in  the  positive  or  negative  direc- 
tion. We  may,  however,  eventually  bring  it  about  through  re- 
verting the  order  of  the  sequence  of  the  end-points  that  the  ex- 
pression 2F  is  always  positive. 

13.  Suppose  that  this  has  been  done.  The  function  2F  is 
to  be  made  a  maximum  under  the  condition  that  the  periphery 
has  a  definite  value  6". 

We  may  write:         51.2-1-5^,3+  . .    .  -|-5„_i,„-h5„.i=6',_ (/3) 

where  5\_i.x  =  V  {x\—x\-^^  -j-  (^\— jx-i)^ (y) 

Form  the  function         G=^2F At &{s^,z  +  Hi^ +5„,i — S), (S) 

and  placing  its  partial  derivatives  =0,  we  have 
dG  ,      {x\ — x\+x  ,  ^>- — x.\^\ 

y  xh  1— r  x-i + ^ + =  o, . 


d  X\  \      5  \+i,  \  5  X_].  X 


(0 
dy\ 

(X=l,  2, . . . .«;  however  for  X^n,  we  must  write:  \  +  l  =  l). 


Take  in  addition  the  equation  (/8)  and  we  have  2n  +  l  equations 
for  the  determination  of  the  2n  +  l  unknown  quantities 

Xi,   Xj,  .  .  .  .  X„,    jt'i,  JK21  •  •  •  -jl^ni     ^• 

14.  In  order  to  come  in  the  simplest  manner  to  the  desired 
result  from  these  equations,  we  adopt  the  following  mode  of  pro- 
cedure.    If  we  write 

£;\  =  {x\ — ;t;x_i)  +  i  (jFx^>'x^i), (0 

then  ^x ,  geometrically  interpreted,  represents  the  length  from  the 
the  point  X — 1  to  the  point  A.  both  in  value  and  in  direction. 
If  further  we  write 

^^=(x\—x\-i)  —  i  (^'x— jj'x-i) (v) 

then  is     ^x-  ^x^-'x-i.X • ^^^ 


of  Functions  of  Several  Variables. 


95 


Multiply  the  first  of  equations  (c)  by  i  and  subtract  from  the  re- 
sult the  second,  then  owing  to  (C)  we  have 


2X^2\ 


.  +  ..•( 


•SX-i.X       -Sx  x^ 


or 


.(0 


Now  multiplying  the  last  two  equations  together,  we  have  from  {&) 


•^x-i,x— -^x.x+i 


and  therefore 

15.     Since  5  ^-^i.  v  is  an  essentially  positive  quantity,  it  follows 

that  5X-:.X=  5x,x+i, {k) 

and  consequently  the  sides  of  the  polygon  are  all  equal  to  one  an- 
other.     Hence  each  side= — ,  and  we  have  from  (i) 

n 


2\+i    e^n  +  s 


-const. 


z\  = —  e  ^ 
n 


z\      em  —  5 
If  we  write 

where  ^x  denotes  the  angle  which  s  x-i,  x  makes  with  the  A'-axis, 
then  e^*^^+-'^^)'=const.,     ' 


or 


(^x+i — ^x=const.; (X) 


that  is,  all  the  angles  of  the  polygon  are  equal  to  one  another, 
and  consequently  the  polygon  is  a  regular  one. 

Weierstrass  thus  showed  that  the  conditions  which  are  had 
from  the  vanishing  of  the  first  derivatives  can  be  satisfied  only 


96  Theory  of  Maxima  avd  Minima 

by  a  regular  polygon;  that  is,  if  there  i.s  a  polygon,  which,  with  a 
given  perimeter  and  a  prescribed  number  -of  sides,  has  a  greatest 
area,  this  polygon  is  necessarily  regular. 

Our  deductions,  however,  have  in  no  manner  revealed  that  a 
maximum  really  exists. 

16.  In  order  to  prove  the  existence  of  a  maximum  we  must 
apply  the  method  given  in  arts.  2-5  of  the  present  chapter.  We 
see  that  a  limit  is  given  for  the  area  of  the  polygon  from  the  fact 
that  the  number  of  sides  and  the  perimeter  are  given;  for,  if  we 
consider  a  square  whose  sides  are  greater  than  the  given  peri- 
meter S,  we  can  lay  each  polygon  with  the  perimeter  S  in  this 
square,  and  in  such  a  way  that  the  end-points  of  the  polygon  do 
not  fall  upon  the  sides  of  the  square.  Hence,  the  area  of  the 
polygon  cannot  be  greater  than  that  of  the  square,  and  conse- 
quently there  must  be  an  upper  limit  for  this  area,  which  may  be 
denoted  by  /%,.  It  may  yet  be  asked  whether  this  limit  can  in 
reality  be  reached  for  a  definite  system  of  values.  The  variables 
Xi,  yi,  X2,  y2,....  x„,  jK„  being  limited  to  this  square,  there  must  be 
among  the  positions  (x^,  y^,  X2,  y2^  ■  ■  •  •  ^ni  ^n)  which  fill  out  the 
square  a  position  {a^,  d^,  a2,  ^2-  •  •  •  -^n^  ^n)  of  such  a  nature,  that  in 
every  neighborhood  of  this  position,  other  positions  exist,  for  which 
the  corresponding  surface  area  F  of  the  polygon  formed  from 
them  comes  as  near  as  we  wish  to  the  itpper  limit.  We  may  as- 
sume that  this  position  is  within  the  square,  since  if  it  lies  by 
chance  on  the  boundary,  then  from  what  has  been  said  above,  it 
is  admissible  to  slide  the  corresponding  polygon  without  altering 
its  shape  and  area  into  the  interior  of  the  square. 

We  assert  that  the  value  of  the  function  F  for  the  position 
(fli,  ^1,  (22,  62,...  .a„,  d„)  must  necessarily  be  equal  to  /%,.  Since,  if 
this  was  not  the  case^  the  inequality  must  also  remain,  if  we  sub- 
ject the  points  a^,  di,  02,  ^2>  •  •  •  •  <^n»  ^n  to  an  infinitely  small  variation; 
and  on  account  of  the  continuity  of  F,  it  would  not  be  possible  in 
the  arbitrary  neighborhood  of  (^i,  d^,...  .a„^5„)  to  give  positions 
for  which  the  corresponding  area  comes  arbitrarily  near  the  upper 
limit  Fo-  This,  however,  contradicts  the  conclusions  previously 
made.  Hence,  all  n  corners  with  a  given  periphery  not  only  ap- 
proach a  definite  limit  with  respect  to  their  inclosed  area  but  this 
limit  is  in  reality  reached.  Since  further  the  necessary  conditions 
for  the  existence  of  a  maximum  have  given  the  regular  polygon  of 


of  Functions  of  Several  Variables.  97 

n  sides  as  the  only  solution,  and  since  we  have  seen  a  maximum 
really  exists,  we  may  with  all  rigor  make  the  conclusion:  That 
polygon,  which,  with  a  given  periphery  and  a  given  number  of 
sides,  contains  the  greatest  area,  is  the  regular  polygon. 

We  have  now  given  what  we  consider  the  general  Weier- 
strassian  theory  of  maxima  and  minima,  and  in  the  sequel  we  shall 
discuss  a  few  special  problems  of  maxima,  and  minima. 

CASES   IN  WHICH   THE   SUBSIDIARY   CONDITIONS   ARE   NOT   TO   BE 
REGARDED  AS  EQUATIONS  BUT  AS  LIMITATIONS, 

17.  Besides  the  problems  already  mentioned^  those  problems 
are  particularly  deserving  of  notice,  in  which  the  conditions  for 
the  variables  are  not  given  in  the  form  of  equations  but  as  re- 
strictions or  limitations. 

For  example,  let  a  point  in  space  and  a  function  which  de- 
pends upon  the  coordinates  of  this  point  be  given.  Further  let 
the  point  be  so  restricted  that  it  always  remains  within  the  in- 
terior of  an  ellipsoid;  then  the  restriction  made  upon  the  point  is 
expressed  through  the  inequality 

—  a^     tr      c^ 

We  have  accordingly  such  limitations  when  a  function  of  the  vari- 
ables is  given  which  cannot  exceed  a  certain  upper  and  a  certain 
lower  limit. 

We  make  such  a  restriction  when  we  assume  that  a  function 
fx  shall  always  lie  between  fixed  limits  a  and  b. 

18.  This  limitation  which  consists  of  two  inequalities 

\A  a<f\<b 

may  be  easily  reduced  to  one. 

For  from  [a]  it  follows  necessarily  that 

b—fx 
and  reciprocally,  if  [/8]  exists,  and  \i  a<^b,  then  /i  must  be  situ- 
ated between  a  and  b  and  consequently  [a]  must  be  true. 

Every  limitation  of  the  kind  given  may  be  analytically  repre- 
sented as  one  single  inequality  of  the  form  [;8]. 


98  Theory  of  Maxima  and  Minima 

19.  We  must  next  find  the  algorithm  for  the  cases  under  con- 
sideration. This  may  be  done  at  once  if  we  consider  that  such 
cases  may  be  reduced  to  those  in  which  occur  equations  of  condi- 
tion. For  this  purpose  we  need  only  establish  the  problem  of  find- 
ing the  maximal  or  minimal  values  of  a  function  whose  variables 
are  subjected  to  certain  conditions  as  follows: 

It  is  required  among  all  systems  of  values  which  satisfy 
the  equations  f\  =  o  (X=l,  2, . . . .  iri),  to  find  those  for  which  F 
is  a  maximum  or  a  minimum. 

By  proposing  the  problem  in  this  manner,  it  is  clear  that  all 
the  variables  x  which  appear  in  the  equations  of  condition  need 
not  necessarily  be  contained  in  the  function. 

Suppose  further  we  have  the  limitation  that 

then,  through  the  introduction  of  a  new  variable  x^^^  we  may  trans- 
form this  limitation  into  an  equation  of  condition.  For,  as  we 
have  to  do  with  only  real  values  of  the  variables,  the  equation 


2 

+1 


denotes  exactly  the  same  thing  as  [y]. 

If  therefore  a  function  F  {x^,  Xj, ^„)  is  to  be  a  maximum 

or  minimum  under  the  limitations 

/,=  0,   fz^O, /m=-0,   f^+i>0,/„,,2>0, /m  +  r>0, 

where  the/'s  are   functions  of  x^,  x^, x„,  then  we  may  solve 

this  problem,  if  instead  of  the  r  last  restrictions,  we  introduce  the 
the  following  limitations: 

The  problem  is  thus  reduced  to  the  one  of  finding  among  the  sys- 
tems of  variables  x^,  X2 x^+,,  those  systems  for  which  F  is  a 

maximum  or  minimum.  , 

20.  Examples  of  this  character  occur  very  frequently  in 
mechanics.  As  an  example,  consider  a  pendulum  which  consists 
of  a  flexible  thread  that  cannot  be  stretched,  then  the  condition 
under  which  the  motion  takes  place  is  not  that  the  material  point 
remains  at  a  constant  distance  from  the  origin,  but  that  the  dis- 


of  Functions  of  Several  Variables. 


99 


tance  cannot  be  greater  than  the  length  of  the  thread.  Such 
problems  are  more  closely  considered  in  the  sequel.  It  will  be 
seen  that  by  means  of  gauss'  principle  all  probletns  of  mechan- 
ics may  be  reduced  to  problem, s  of  m.axim^a  and  minim,a. 

gauss'  principle. 

21.  For  the  sake  of  what  follows,  we  shall  give  a  short  ac- 
count of  this  principle:  Consider  the  motion  of  a  system  of  points 
whose  masses  are  m.^,  m-2, . . . .  ;w„ .  Let  the  motions  of  the  points 
be  limited  or  restricted  in  any  manner,  and  suppose  that  the  sys- 
tem moves  under  the  influence  of  forces  that  act  continuously. 
For  a  definite  time,  let  the  positions  of  the  points  and  the  com- 
ponents of  velocity  both  in  direction  and  magnitude  be  deter- 
mined. The  manner  in  which  the  motion  takes  place  from  this 
period  on  is  determined  through  Gauss'  Principle: 

Let  Ai,  A2,. . .  .A„,  be  the  positions  of  the  points  at  the 
moment  first  considered; 

Bi,  B2,....B„,  the  positions  which  the  points  can  take  after 
the  lapse  of  an  infinitely  small  time  t,  if  the  motions  of  these 
points  are  free; 

Ci,  Cj, . . . .  C„,  the  positions  in  which  these  points  really  are 
after  the  lapse  of  the  same  time  t;  and  finally  let 

C/,  Cj', .  . .  .  C„'  be  the  positions  which  the  points  may  also 
possibly  have  assumed  after  the  time  t,  when  the  conditions  are 
fulfilled. 

If  we  form 


^  m^v  By  Cv      and  ^  ^^^  Bv  Cj  , 


'-I 


"=1 


it  follows  from  Gauss'  Principle,  that  from  t=c?  up  to  a  definite 
value  of  T,  the  condition 


[1] 


2  ^>'  BpCy       <    2  ^''  B'  ^^ 


"=1 


"=1 


is  always  satisfied,  that  is,    ^  ^•'  ^^  ^^     "lust  always  be  a  mini- 


mum. 


"=1 


100  Theory  of  Mwsiima  and  Minima 

22.  In  order  to  make  rigorous  conclusions  from  Gauss'  Prin- 
ciple, which  was  briefly  sketched  in  the  preceding  article,  we  shall 
give  a  more  analytic  formulation  of  it :  For  this  purpose  we  de- 
note the  coordinates  of  Ay  by  Xy ,  yy,  Zy,  the  components  of  the 
velocity  of  A  by  x\  y'^,  z'^,  the  components  of  the  force  acting 
upon  Ay  by  Xy,    Yy,  Zy.       The  coordinates  of  By  are  therefore 

Xv  +  TX'y    +    —Xy,yy+    T  /  y    +     —    K^  ,       Zy    +    TZy    +    —Zy    \ 

and  from  Taylor's  theorem  the  coordinates  of  d-  are 

_2  _2  2 

consequently  we  have  * 

[2]2^v^^;^=2^''{«-^v)'+(X'-n)'+«-^v)^}^V.... 


Instead  of  x'^,   however,  (see  preceding  art.)   other  values  may 
possibly  enter,  say  x''^^^, ,  so  that  we  have 


[3]  ^myByC\^ 


y=\ 


It  follows  from  Gauss'  Principle  that  the  difference  of  the   sums 
[2]  and  [3]  must  always  be  positive. 
Hence 

[4]     c.>2^''{2[^''«-^,)  +  ^,(X'-n)+C,«-Z/)]  + 


"=1 


that  is,  the  quantities  x'^  ,  y'J ,  z'J  must  be  such  that  the  sum  [2] 
is  a  minimum. 


of  Functions  of  Several  Variables.  101 

Hence  among  all  the  xi^ ,  y'^ ,  z'l  which  are  associated  with 
the  conditions  of  motion,  we  must  seek  those  which  make  [2]  a 
minimum. 

23.  We  have  reached  our  proposed  object,  if  we  can  show 
that  the  ordinary  equations  of  mechanics  may  be  derived  from 
Gauss'  Principle. 

If  there  are  no  equations  of  condition  present,  then  clearly  [2] 
is  only  a  minimum  when 

If,  however,  we  have  equations  of  condition,  for  example, /"(;i;, 
y,  s)=^o,  then  these  must  hold  true  throughout  the  whole  motion. 
They  may  therefore  be  differentiated.     We  have  in  this  way  equa- 
tions in  ^^.    -^   and    -^  •     Differentiate  again  and  we  have 
at       at  at 

equations  in  x'^,  y'^  and  s'^ . 

Hence,  in  conformity  with  the  rules  that  have  been  hitherto 
found  for  the  theory  of  maxima  and  minima,  the  quantities 
^v'  X''  K  are  to  be  so  determined  that  the  derived  equations  of 
condition  are  satisfied,  while  at  the  same  time  [2]  becomes  a 
minimum.  But  in  this  case  also,  as  is  easily  shown,  we  are  led  to 
the  usual  differential  equations  of  mechanics. 

TWO   APPLICATIONS   OP   THE   THEORY   OP   MAXIMA    AND    MINIMA 
TO   REALMS   THAT   SEEM   DISTANT   PROM   IT. 

I.  Cauchy's  proof  of  the  existence  of  the  roots  of  algebraic 
equations. 

24.  An  interesting  application  of  the  theory  of  maxima  and 
minima  (cf.  art.  5)  is  Cauchy's  existence-proof  of  the  roots  of  an 
algebraic  equation: 

Let/C^)  be  an  integral  rational  function  of  a  real  or  com- 
plex variable  z.     The  function  becomes,  when  z  is  put  ~-=^  x  -|-  i  y, 

f(z)=(l)(x,y)  +  iy\i  (x,y), 
where  (f>  and  \jf  are  real  functions  of  the  real  variables  x  and  y. 

Hence  <fP(x,y)+^(x,y)  is  always  a  positive  quantity.  The 
variables  ;i;  and  j>^  may  be  arbitrarily  great;  but,  if  x^+y^~-the 
square  of  the  distance  of  the  positiion  (x,  y)  from  the  origin  —  is 
infinitely  large,  then  (f>^  +  ^  is  infinitely  large. 


102  Theory  of  Maxima  and  Minima 

Draw  a  circle  about  the  orij^in  ;  then  the  function  4'^  +  -^  is  de- 
fined both  for  the  interior  and  the  boundary  of  this  circle.  We 
may  choose  the  radius  of  the  circle  so  great,  that  the  function  for 
every  point  of  the  circumference  is  greater  than  it  is  for  any  arbi- 
trary point  within  the  interior.  The  function  <^^  +  i/»^  must  also 
have  a  lower  limit,  and  since  it  is  a  continuous  function,  there 
must  be  a  position  within  the  interior  of  the  circle  where  (^^4-\|(^is 
a  minimum.  Let  this  be  the  case  for  the  point  {x^t^y^)-  It  is  then 
easy  to  show  that  <f>^  +  rji^  can  have  no  other  value  for  (x^,,  ^o)  than 
zero.  /"(^)  is  also  =o  for  {x^,  jj'o);  and  therefore  ^0=^0  + ^J^o  is  a 
root  of  the  equation /"(^)  =  o. 

II.  Proof  of  a  theorem  in  the  Theory  of  Functions.  The 
reversion  of  series. 

25.  The  theorem  which  was  stated  in  arts.  14-15  of  Chap- 
ter I.,  and  of  which  an  application  has  already  been  made  in  arts. 
4-6  of  Chapter  III.,  is  of  great  importance  in  the  Calculus  of 
Variation. 

If  the  n  equations  exist  between  the  variables  x^,  Xj,  . . .  .x„, 
7u  yv--x„- 

yi=  a.xx  Xx  +  «i2  ^2  4- +  «1„  ^n  +  Ai' 

\  yn-=  a^nXi  +  a„ 


[1] 


t-„2  X2 


+ +  a„„  x^  +  A„, 


where  the  coefficients  on  th^  right-hand  side  are  given  finite 
quantities,  and  the  j\^s  are  power-series  in  the  x's  of  such  a 
nature,  that  each  single  term  is  higher  than  the  first  dimension, 
and  if  the  series  on  the  right-hand  side  are  convergent  and  the 
determinant  of  the  nth  order  of  the  linear  functions  of  the  x's, 
which  appear  in  [1], 


[2]  A  = 


^11'     ^12'      '^lii 

^21'     '^22>      '^211 

^  11  1     1      a  „7    ,        a  nn 


of  Functions  of  Several  Variables. 


10.^ 


is  different  from  zero,  then,  reciprocally ,  the  x's  may  also  be 
expressed  through  convergent  series  of  the  n  quantities  y, 
which  identically  satisfy  the  equations  [1]. 

26.  As  an  algorithm  for  the  representation  of  the  series  for 
the  ;c's,  we  made  use  of  the  following  methods  (see  arts.  14  and 
15,  Chapt.  I.): 

We  solved  the  equations  [1]  linearly  by  bringing  the  terms 
of  the  higher  powers  of  the  x'^  on  the  left-hand  side,  and  thus  had 


X. 


A 


f(^.-X.)+-^"(^-X)^  ■•+^(^..-X.)- 


x„ 


A, 


A 


<-X!).t(--XO 


+■■ 


A^ 
A 


■(^"-X") ; 


where  Axp.  denotes  the  corresponding  first-minor  of  a\^  in  [2]. 
It  is  seen  that  in  ofeneral 


M^ii 


[3] 


x\^ 


A\^ 
A 


"^^  ^V^,  L"^l'  '''2! 


.Tm 


■^^' 


A 


^=1 


^=1 


We  shall  therefore  have  a  first  approximation  to  the  result,  if  we 
consider  only  the  terms  on  the  right-hand  side  of  [3],  which  are 
of  the  first  dimension.  A  second  approximation  is  reached,  if  we 
substitute  in  the  right-hand  side  of  [3]  the  first  approximations 
already  found,  and  reduce  ever3'thing  to  terms  of  the  second 
dimension  inclusive.  Continuing  with  the  second  approximations 
that  have  been  found,  substitute  them  in  [3],  and  neglecting  all 
terms  above  the  third  dimension,  and  we  have  the  third  approxi- 
mation, etc.;  we  may  thus  derive  the  ;i;'s  to  any  degree  of  exact- 
ness required. 

Since  A  is  found  in  all  the  denominators,  the  development 
converges  the  more  rapidly  the  greater  A  is. 

27.  In  what  follows  we  shall  assume  that  the  quantities  on 
the  right-hand  side  of  [1]  are  all  real  and  that  we  may  write 


L 


=  Ax  2  + Ax, 


■A\^+..:.  . 
(^=1,2, 


n) 


104  Theoiy  of  Maxima  and  Minima 

where  A\i  is  a  homogeneous  function  of  the  /th  degree  in  ;c„  X2, . 
. .  x^  and  consequently  : 


^^  O  Xi  0X2 


+  3x,^+3x2  ^^^^ 


d  Xi  d  X, 


2 


+  Ax,  ^+  4X2  ^^'' 


+  2x„ 

a^\2 
dx„ 

+  3x, 

1 

dAx, 
dx„ 

+  Ax„ 

dAx, 
0  -. 

d  Xi  d  X2    ' 

-1- ; 

(\=l,2,....n) 

or,  A^^^^i  Am  +  ^2Am-1- f^nAx"  • 

The  quantities     Av    (^=^»  2, .  . .  .n;  )u,  =  l,  2, .  .  •  .n)  are  continu- 
ous functions  of  the  x's  which  become  infinitely  small  with  the  ;c's. 
The  system  of  equations    [1]   may  be  then  brought  to  the 
form  [1] 

(X=l,2,..../^) 


The  theorem  of  art.  25  in  this  modified  form,  may  be  expressed  as 
follows : 

(1)  //  is  always  possible  so  to  fix  for  the  variables  x,,  X2,...  .x^ 
and y\,  y2,  ■  ■  ■  -y^,  limits  g,,  g2,---  -gn  (^'>^d  h,,  ^2>  ■  •  •  •  ^^n  that  for 
every  systein  of  the  y's  for  which  \y\\  <C  Ax  there  exists  ONE  5^5- 
tein  of  the  x's,for  which  !;t;x|<C^x,  and  in  such  a  way  that  the 
equations  [la]  are  satisfied. 

(2)  The  solution  of  the  equations  [1<^]  has  a  similar  form  as 
the  equations  [li?]  themselves,  viz. : 

M  -x=2  (^+Yx.)>. 


M=l 


(X=l,  2.....;^) 


of  Functions  of  Several  Variables. 


105 


where  the  \\ii  are  cofftinuous  function  of  the  y'  s  which  become 
infinitely  small  with  these  quantities. 

To  prove  this  theorem  we  make  use  of  the  theory  of  maxima 
and  minima. 

28.  If  we  give  to  the  y\  the  value  zero,  the  equations  [1«] 
can  only  be  satisfied  if  their  determinant  vanishes,  that  is,  when 


[4] 


a\)x 


\x^  I  =^' 

(X,  ^^=1,  2 n) 


except  for  the  case  where  the  ;t;'s  vanish. 

For  sufficiently  small  values  of  the  Jt's,  the  determinant  [4]  is 
not  very  different  from  the  determinant  [2].  We  may  therefore 
determine  limits  g  for  the  x'^,  so  that  [4]  cannot  be  zero  unless  [2] 
is  also  zero.  A=.o  is,  however,  by  hypothesis  excluded.  Accord- 
ingly the  y  s  can  only  be  zero  in  \\a\  when  all  the  ;p's  vanish. 
The  ;c's  are  thus  confined  within  fixed  limits  which  may  be  re- 
garded as  the  boundaries  of  a  definite  realm. 

29.     Again  we  write 


M=« 


(\=1,2.....^) 


/»=1 


and  consider  the  function 


[6] 


x^l     -''  X    ^'^'^  '''2'  •  •  •  •  ^ni  )• 


In  6*  we  will  write  for  the  ;r's  all  the  systems  of  values,  where  at 
least  one  x  lies  on  the  boundary  of  the  realm  in  question.  We  re- 
gard as  the  realm  of  the  x'^  the  totality  of  the  x'%  for  which 


a\ 


M  +  y^  \  ^      is  only  zero,  when 
(X,/*=l,  2,....«) 


a\i 


^o\ 


it  follows  then  that  [4]  is  not  zero,  since  \a\y\  is  by  hypothesis 
different  from  zero. 


R 


106 


Theory  of  Mai^hna  and  Minima 


In  order  to  make  this  clear,  it  may  be  mentioned  that  as  the 
boundaries  of  the  realm,  we  must  consider  the  totality  of  the  ;ir's 
where  at  least  one  of  the  ;t:'s  reaches  its  limit. 

For  the  limits^  that  is,  when  one  of  the  x''&  reaches  its  limit, 
there  is  no  system  of  values  of  the  x'^  for  which  the  function  [6] 
vanishes^  since  the  function  can  (as  follows  from  definition  [5]  of 
the  F\  and  the  considerations  of  art.  28)  only  vanish,  if  all  the  jk's 
and  consequently  all  the  x'%  vanish. 

There  is  then  a  lower  limit  G  which  is  difFerent  from  zero  for 
the  values  of  [6],  which  correspond  to  a  system  of  values  {x-^,  Xj, . . 
.  .x„)  oi  the  limits. 

30.  We  come  next  to  the  determination  of  the  limiting  val- 
ues of  the  y  s.     For  this  purpose  we  consider 


[7] 


2( 

X=l 


-/* X  \Xi,  Xj,  ....  x^ ) - 


y\ 


y 


If  we  ascribe  definite  values  to  the  y s^  then  there  is  for  the  values 
[7]  in  the  realm  of  the  x's,  a  system  for  which  [7]  is  a  minimum. 

We  wish  to  show  that  this  system  of  values  of  the  x's  does 
not  lie  upon  the  boundary  of  the  realm.  We  prove  this  by  show- 
ing that  there  is  a  point  within  the  realm,  where  the  expression 
[7]  has  a  smaller  value  than  it  has  on  the  boundary : 

The  expression  [7]  may  be  written 


x=« 


x=» 


^{FK-yxy=.S—2^Fx_yx    1-2 


^ 


X=i 


Since  i    S  is  at  all  events  greater  than  /"'a.  and  consequently 


Fk 


)    S 


=  <1, 


it  follows  that 


li=n 


f-=n 


/*=« 


/*-i 


Ai=i 


M=l 


ii=n 


H=^n 


M=l 


M=l 


of  Functions  of  Several  Variables.  107 

where  the  A's  are  the  limits  of  the  ^s.     From  this  it  results  that 

2  ^F,  -y,  Y>  s-  2 1  Ts-"  2  ^'^  +  2  yl ' 

Ml  M-^l  Ml 

and,  consequently,  for  a  greater  reason 

[8]  2  (^'m->'m)^>^V-2i^1^2^'^- 

M-l  M-1 

The  limits  have  to  be  so  chosen  that  the  right-hand  side  of 
[8]  is  positive.  This  choice  can  be  made  so  that  the  expression 
on  the  right-hand  side  for  a  system  of  x''^,  which  belongs  to  the 
boundary,  does  not  become  arbitrarily  small,  but  always  remains 
greater  than  a  certain  lower  limit  (see  the  preceding  article). 

The  expression,  however,  on  the  interior  of  the  realm  of  the 
;i:'s  may  be  arbitrarily  small,  viz.:  when  Xi=X2  =  .  . .  .  =x„^o. 

For  this  system  of  values  the  left-hand  side  of  [8]  is  equal  to 

^=11 

M=l 

We  have  therefore  found  the  following  result:  We  can  give 
limits  g  to  the  variables  x,  and  to  the  y' s  the  limits  h  in  such 
a  way  that  the  expression  [7]  for  systems  of  values  of  the  x's, 
which  belong  to  the  boundary  of  the  realm,  is  always  greater 
than  it  is  for  the  zero  position  {^x-^=Xi=  ....  =x^—o^. 

Hence  the  position  for  which  the  expression  [7]  ^5  a  tnini- 
■mum  mttst  necessarily  lie  within  the  realm  of  the  x's;  and  we 
may  be  certain  that  within  the  realm  of  the  x's  there  is  a  posi- 
tion where  [7]  has  its  smallest  value. 

31.  In  order  to  find  the  minimal  position  of  [7]  which  was 
shown  to  exist  in  the  previous  article,  we  must,  following  the 
rules  laid  down  in  Chapt.  III.,  differentiate  the  function  [7]  and 
put  the  first  partial  derivatives =<?. 

This  gives 


[9]  ^{F.-y.) 


=  0. 


dxp. 
(/t=l,2,....n) 


I 


108  Theory  of  Maxima  and  Minima 

These  n  equations  can,  in  case  the  determinant 

[10] 


dF, 


(v,  (1=1,2, n) 

is  different  from  zero,  only  exist,  if  the  quantities  within  the 
brackets  vanish. 

[10]  is  identical  with  the  determinant  [4],  and  (see  art.  28) 
it  may  be  always  brought  about  through  suitable  choice  of  the 
limits  g  of  the  x's  that  [4]  is  different  from  zero,  if  only  the  de- 
terminant [2],  as  by  hypothesis  is  the  case,  is  different  from  zero. 

Hence  the  equation  [9]  can  only  be  satisfied  if 

[la]  or  [5]  }'v  =  Fp  (xi,  x^,....  x^. 

We  have  therefore  found  that,  since  there  is  certainly  a  sys- 

''=n 

tern  of  values  of  the  x's  for  which  the  function  ^  i^yv — Fv  )^  is 

"=1 
a  minimum,  there  must  also  be  a  system,  within  the  realm  of 

the  x's  for  which  the  equations  \la\  are  satisfied,  if  to  the  y's 

definite  values  in  their  realm  are  arbitrarily  given. 

32.  We  must  further  see  whether  within  the  fixed  realm 
there  is  one  or  several  systems  of  values  of  the  x''^,  that  satisfy 
the  equations  [la]  with  prescribed  values  of  the  ys  which  lie 
within  definite  limits. 

In  order  to  establish  this,  we  assume  that  {x\,  x\, . . .  .y„)  is 
a  second  system  of  values  that  satisfy  the  equations  [la];  we 
must  then  have  the  equations 

[11]  Fv  (x\,  x'2, x'„)—Fv  (x„  X2 x„)  =  o. 

{v=l,Z....n) 

Developing  by  Taylor's  theorem,  we  have,  when  we  consider  only 
terms  of  the  first  dimension 


[11a]  2(^'^-^^)]lf +  X. 


p=i 


)  dXp  ^  ^"'^  f 

(v=l,2,....n) 


of  Functions  of  Several  Variables.  109 

The  X  „p  are  functions  which  depend   upon  the  x'^'s,  and  ;«;'s  and 

vanish  with  these  quantities. 

We  will  determine  the  n  unknown  quantities  x\ — x-^,  x'^ — Xj, . . 
.  .x'a — Xa  from  the  n  linear  equations  {11(2]  . 

For  small  values  of  x  and  x'  the  determinant 

M  T^r-^A- 1 

will  be  little  different  from  the  determinant  [10]. 

We  may  therefore  make  the  limits  g  of  the  ;i:'s  so  small  that 
[12]  is  different  from  zero  for  all  the  x's  and  x"s  which  belong  to 
the  realm. 

But  then  the  equations  [11«]  are  only  satisfied  for 

x'p=Xp      (p=l,  2, n); 

that  is,  there  exists  within  the  realm  in  question  no  second  sys- 
tem of  the  x's  which  satisfies  the  equations  [l^z]. 

We  have  therefore  come  to  the  following  result : 

It  is  possible  so  to  determine  the  litnits  g  and  h  that  with 
every  arbitrary  system,  of  the  y's  in  which  each  single  variable 
does  not  exceed  its  definite  limiting  value,  the  given  equations 
[1«]  are  satisfied  by  ONE  and  only  one  syste-tn  of  the  x's,  in 
which  these  quantities  likewise  do  not  exceed  their  limits. 

The  first  part  of  the  theorem  given  in  art.  26  is  thus  proved. 

Remark. — We  have  assumed  that  we  have  to  do  only  with 
real  quantities.  The  discussion,  however,  is  not  restricted  to  such 
quantities,  as  it  is  easy  to  prove  that  the  same  developments  may 
be  also  made  for  complex  variables. 

33.     The  values  of  the  x'^  which  were  had  from  the  equations 


[1«]  >'''=2  («"?  +  A-p)^'' 

(i'=:l,2,....n) 


110  Theory  of  Maxima  and  Minima 

may  be  derived  in  the  manner  given  in  art.  26: 
If  we  write 


«.'/'+ A-p   =^'' 

(v,  p=l,  2,. 

...«) 

the  linear  solution  of  the  equations  \la\  is 

x                   ^.^-^li^v.. 

(P=l,2,... 

.^) 

where  A'vp  denotes  the  corresponding  subminor  of  A' .     Now  A' 
is  a  definite  quantity  which  lies  within  certain  finite  limits  ;  the 

same  is  also  true  of  — , .     ^';.  p  is  found  in  a  similar  manner.     Hence 

A 

A' 
the  quantities  — ~~~  are  finite  quantities  which  lie  between  defin- 
ite limits  ;  and  therefore,  if  the  y s  become  infinitely  small,  the  ;i;'s 
will  also  become  infinitely  small ;  that  is,  those  systems  of  values 
of  the  x'%,  which  satisfy  the  equations  [1]  under  the  named  con- 
ditions, are — as  has  also  been  shown  in  art.  28— so  formed  that 
they  become  infinitely  small  with  the  jv's. 

We  can  now  show  that  the  ;t;'s  are  continuous  functions  of 
the  y  s. 

Let  (^1,  ^2^. . .  .^„)  be  a  definite  system  of  values  of  the  jf's 
and  let  the  system  (^i,  a^, a„)  of  the  ;i;'s  correspond  to  this  sys- 
tem of  the  ys. 

If  we  then  write 
■     [13]  (jx=^x+,x 

the  system  of  equations  [la]  or  [1]  goes  into 

(X=l,2 n) 


of  Functions  of  Several  Variables. 


Ill 


or 


(X=l,2,....«) 
Developing  this  expression  according  to  powers  of  the  ^'s,  we  have 


/i=« 


[Ic] 


TJX  =  2  ^'^-^  ^-^  ' 

(X=l,  2,. 


/*-=! 


.«) 


where  the  y4'x/x  are  functions  of  the  i?'s  and  f's.  If  now  the  f's 
are  infinite!}'  small,  we  may  limit  the  A'\fL  to  the  first  derivatives 
oiFx.  In  this  case  we  denote  the  coefficients  of  [Ic]  by  Ax/x,  so 
that 

A\^i  =  -^—  ior{xi=-ai,  X2=a2, ^n=«n), 


d  Xpi 


(X,  fc=l,  2,....n) 


and  the  determinant  of  the  equations  [1]  goes  into 
dFx 


dxn 


for  (xi—a^,  Xi=a2, . . .  .x^^aa)- 
(X,,i=l,2,..    .n) 


If  the  x'^  lie  within  definite  limits,  this  determinant  remains 
always  above  a  definite  limit.  We  may  therefore  say  that  the 
determinant  has  a  value  different  from  zero.  Consequently  the 
condition  that  the  equations  [Ic]  may  be  solved,  is  satisfied,  and 
it  is  seen  that  infinitely  small  values  of  the  |^'s  must  correspond  to 
infinitely  small  values  of  the  •rj's. 

This  means  nothing  more  than  that  the  functions  of  the  x'^ 
are  continuous  functions  of  the  ys, 

34.  Our  investigations  are  true  under  the  assumption  that 
the  functions  F\  are  continuous,  that  their  first  derivatives  exist 
and  likewise  are  continuous  within  certain  limits.  We  need  know 
absolutely  nothing  about  the  second  derivatives. 

Of  the  x'^,  of  which  we  already  know  that  they  exist  as 
functions  of  the  ^s  and  vary  in  a  continuous  manner  with  them, 
we  may  now  likewise  prove  that  they,  considered  as  functions  of 
the  ys,  have  derivatives  which  are  continuous  functions  of  the  y&. 


112  Theory  of  Maxima  and  Minima 

We  have  then  proved  that  the  x'^,  are  such  functions  of  the 
y^  as  the  ys  are  of  the  x^. 

The  proof  in  question  may  be  derived  from  the  following  con- 
siderations:— If  from  [Ic]  we  express  the  ^'s  in  terms  of  the  t^'s, 
we  have 

()a=l,2,....^) 

The      ^^     are  continuous  functions  of  the  f 's,  and  the  ^'s  are 
A 

continuous  functions  of  the  tj's.  Hence  —^  may  be  represented 
as  continnous  functions  the  tj's. 

If  the  Tj's  become  infinitely  small,  then  the  ^'s  become  infin- 
itely small,  and  we  have  limits  for  ^-^• 

If  we  have  in  general  a  function  of  the  n  variables  x-^,  x-i, . . 
.  .x„,  and  if  we  consider  the  difference 

it  is  seen  that  it  may  be  written  in  the  form 


2(^x+^x)^^ 


where  the  H\  depend  upon  the  ^'s  and  become  infinitely  small  with 
these  quantities,  and  the  b\  are  the  partial  differential  quotients 
of  f  with  respect  to  x\  for  the  system  of  values  {a^,  Uj, . . .  .»„) . 
From  this  it  results  the  ;i;'s  are  not  only  continuous  functions  of 
the  jf's,  but  also  that  the  derivatives  of  the  first  order  of  the 
functions  exist. 


We  have  indeed  the  derivatives  of  the  first  order,  if  in  the 

•essions  — -^  we  write  the  ^'s  equal  to  zero. 

The  quantities  — ^  ,  however,  become  then  in  accordance  with 


of  Functions  of  Several  Variables. 


113 


[Ic],  the  quantities  which  we  would  have  in   [Ic],  if  we  had  at 
first  written  A\^  instead  of  A'xi,.. 

But  the  quantities  A\iL  are  continuous  functions  of  aj,  ^2, .. 
.  .a^.  We  may  therefore  say  that  the  differential  quotients  — ^ 
are  continuous  functions  of  the  variables  x. 


35.  For  the  complete  solution  of  the  second  part  of  the 
theorem  in  art.  26,  we  have  yet  to  show  that  the  expressions  [3^] 
may  be  reduced  to  the  form  \Za\\ 

For  this  purpose  we  must  bring  the  quantities  — ^  in  [35] 
(art.  32)  to  the  form 


where  b\u.  is  the  value  of 


A' 

A\y. 

A' 


when   all   the    ;i;'s    are   equal    to 


zero.       \\y.  is  a  function  of  the  ;«;'s,  but  the  ;i;'s  are  functions  of 

theys,  so  that  \\^  is  a  function  of  thejv's  which  vanishes  when 
they  vanish. 

We  may  therefore  in  reality  write  [35]  in  the  form  [3a] 

x=« 

^M-=2  ('^^''  +  Yxm)  y>^- 

(/i-l,  2,....«) 


[3«] 


X=l 


36.  There  may  arise  cases  in  which  we  know  nothing  further 
of  the  functions  F\ — as  was  assumed  in  art.  26 — than  that  they 
are  real  continuous  functions. 

We  cannot  then  conclude,  for  example,  that  the  x'^  may  be 
developed  in  powers  of  theys;  but  we  may  reduce  the  equations 
to  the  form   [3c]   and  show  that  the  equations  [la]   are  solvable. 

The  theorem  which  has  been  proved  is  of  great  importance 


114  Theory  oj  Maxima  and  Minima 

when  applied  to  special  cases,  even  for  elementary  investigations. 

If,  for  example,  the  equation /"(;?;,  ;k)=o  is  given,  then  it  is 
taught  in  the  differential  calculus  hovi^  we  can  find  the  derivative 
of  y  considered  as  a  function  of  x. 

If  we  assume  that  the  variables  x  and  y  are  limited  to  a 
special  realm,  where  the  two  derivatives  with  respect  to  x  and  y 
do  not  vanish,  and  therefore  the  curve  /{x,y)=o  has  no  singu- 
lar points;  and,  if  the  equation  is  satisfied  by  the  system  (^oijo). 
we  may  write  x=X(,  +  ^,  y^y^j^-q.  We  have  then  /{xQ  +  ^,yo+'n) 
=0,  and  we  may  prove  with  the  aid  of  the  theorem  in  art.  27  that 
Tj  is  a  continuous  function  of  ^  and  has  a  first  derivative.  Not 
before  this  has  been  done,  have  we  a  right  to  differentiate  and 
proceed  according  to  the  ordinary  rules  of  the  differential  calculus. 


^     OF  THE 

UNIVERSITY 


OF 

JFORti^ 


IN  PRESS :     Lectures  on  the 


CALCULUS  OF  VARIATIONS. 


By  Professor  Harris  Hancock. 


y 


m 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
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a&m  DEPT. 


20-' 


ceb'^*--' 


22Way'64DVI/ 


RECD  UD 


RECD  LD 

NOV  18  1957 


Rfe.w  ^  LD     1 1 


JAN  2  6  ^OGG  ^  S 


LD  21-100m-7,'52(A2528sl6)476 


I  ^  "on 


